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| − | Friedman's work
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| − | Weak tree function
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| − | TREE function
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| − | Kruskal's tree theorem
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
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| − |
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| − | History
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| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
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| − |
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| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
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| − | TREE
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| − | (
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| − | 3
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| − | )
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| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
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| − |
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| − | Statement
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| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
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| − |
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| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
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| − |
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| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
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| − | T
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| − | 1
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| − | ≤
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| − | T
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| − | 2
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| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
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| − |
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| − | For all vertices v of T1, the label of v precedes the label of
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)};
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| − | If w is any successor of v in T1, then
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| − | F
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| − | (
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| − | w
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| − | )
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| − | {\displaystyle F(w)} is a successor of
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)}; and
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| − | If w1, w2 are any two distinct immediate successors of v, then the path from
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| − | F
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| − | (
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| − | w
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| − | 1
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| − | )
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| − | {\displaystyle F(w_{1})} to
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| − | F
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| − | (
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| − | w
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| − | 2
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| − | )
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| − | {\displaystyle F(w_{2})} in T2 contains
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)}.
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| − | Kruskal's tree theorem then states:
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| − |
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| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
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| − | i
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| − | <
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| − | j
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| − | {\displaystyle i<j} so that
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}}.)
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| − |
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| − | Friedman's work
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| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
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| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
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| − | 1-CA0.
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| − |
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| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
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| − |
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| − | Weak tree function
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| − | Suppose that
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is the statement:
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| − |
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| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
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| − | i
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| − | +
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| − | n
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| − | {\displaystyle i+n} vertices, then
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}} for some
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| − | i
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| − | <
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| − | j
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| − | {\displaystyle i<j}.
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| − | All the statements
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
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| − |
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| − | Define
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| − | tree
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
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| − |
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| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
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| − | i
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| − | +
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| − | n
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| − | {\displaystyle i+n} vertices, such that
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
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| − | i
| |
| − | <
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| − | j
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| − | ≤
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| − | m
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| − | {\displaystyle i<j\leq m}.
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| − | It is known that
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| − | tree
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| − | (
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| − | 1
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| − | )
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| − | =
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| − | 2
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| − | {\displaystyle {\text{tree}}(1)=2},
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| − | tree
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| − | (
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| − | 2
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| − | )
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| − | =
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| − | 5
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| − | {\displaystyle {\text{tree}}(2)=5},
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| − | tree
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| − | (
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| − | 3
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| − | )
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| − | ≥
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| − | 844
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| − | ,
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| − | 424
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| − | ,
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| − | 930
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| − | ,
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| − | 131
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| − | ,
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| − | 960
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| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
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| − | tree
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| − | (
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| − | 4
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| − | )
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| − | ≫
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| − | g
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| − | 64
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| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
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| − | g
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| − | 64
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| − | {\displaystyle g_{64}} is Graham's number), and
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| − | TREE
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| − | (
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| − | 3
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| − | )
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| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
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| − | t
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| − | r
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| − | e
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| − | e
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| − | t
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| − | r
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| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
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| − | e
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| − | e
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| − | 8
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| − | (
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| − | 7
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| − | )
| |
| − | (
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| − | 7
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| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
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| − | 7
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| − | )
| |
| − | (
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| − | 7
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| − | )
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| − | .
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| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
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| − |
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| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
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| − |
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| − | TREE function
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| − | Sequence of trees where each node is colored either green, red, blue
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| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
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| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
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| − | TREE
| |
| − | (
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| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
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| − |
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| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
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| − | j
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| − | ≤
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| − | m
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| − | {\displaystyle i<j\leq m}.
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| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
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| − | 1
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| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
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| − | 3
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| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
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| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
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| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
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| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
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| − | A
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| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
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| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
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| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | Categories: Mathematical logicOrder theoryTheorems in discrete mathematicsTrees (graph theory)Wellfoundedness
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
| |
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| − | Categories: Mathematical logicOrder theoryTheorems in discrete mathematicsTrees (graph theory)Wellfoundedness
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
| |
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | TREE function
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| − | Kruskal's tree theorem
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
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| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
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| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
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| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
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| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
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| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | TREE function
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| − | Kruskal's tree theorem
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
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| − |
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| − | History
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| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
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| − |
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| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
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| − | TREE
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| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
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| − |
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| − | Statement
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| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
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| − |
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| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
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| − |
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| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
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| − | T
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| − | 1
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| − | ≤
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| − | T
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| − | 2
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| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
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| − |
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| − | For all vertices v of T1, the label of v precedes the label of
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| − | F
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| − | (
| |
| − | v
| |
| − | )
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| − | {\displaystyle F(v)};
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| − | If w is any successor of v in T1, then
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| − | F
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| − | (
| |
| − | w
| |
| − | )
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| − | {\displaystyle F(w)} is a successor of
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| − | F
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| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
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| − | If w1, w2 are any two distinct immediate successors of v, then the path from
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| − | F
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| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
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| − | F
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| − | (
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| − | w
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| − | 2
| |
| − | )
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| − | {\displaystyle F(w_{2})} in T2 contains
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| − | F
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| − | (
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| − | v
| |
| − | )
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| − | {\displaystyle F(v)}.
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| − | Kruskal's tree theorem then states:
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| − |
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| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
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| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
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| − | {\displaystyle T_{i}\leq T_{j}}.)
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| − |
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| − | Friedman's work
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| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
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| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
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| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
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| − |
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| − | Weak tree function
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| − | Suppose that
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| − | P
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| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
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| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
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| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
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| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
| |
| − | vte
| |
| − | Large numbers
| |
| − | vte
| |
| − | Order theory
| |
| − | Categories: Mathematical logicOrder theoryTheorems in discrete mathematicsTrees (graph theory)Wellfoundedness
| |
| − | This page was last edited on 21 October 2024, at 22:33 (UTC).
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| − | Friedman's work
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| − | Weak tree function
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| − | TREE function
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| − | Kruskal's tree theorem
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
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| − |
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| − | History
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| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
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| − |
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| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
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| − | TREE
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| − | (
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| − | 3
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| − | )
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| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
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| − |
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| − | Statement
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| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
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| − |
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| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
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| − |
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| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
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| − | T
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| − | 1
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| − | ≤
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| − | T
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| − | 2
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| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
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| − |
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| − | For all vertices v of T1, the label of v precedes the label of
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)};
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| − | If w is any successor of v in T1, then
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| − | F
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| − | (
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| − | w
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| − | )
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| − | {\displaystyle F(w)} is a successor of
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)}; and
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| − | If w1, w2 are any two distinct immediate successors of v, then the path from
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| − | F
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| − | (
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| − | w
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| − | 1
| |
| − | )
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| − | {\displaystyle F(w_{1})} to
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| − | F
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| − | (
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| − | w
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| − | 2
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| − | )
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| − | {\displaystyle F(w_{2})} in T2 contains
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)}.
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| − | Kruskal's tree theorem then states:
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| − |
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| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
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| − | i
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| − | <
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| − | j
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| − | {\displaystyle i<j} so that
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}}.)
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| − |
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| − | Friedman's work
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| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
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| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
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| − | 1-CA0.
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| − |
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| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
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| − |
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| − | Weak tree function
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| − | Suppose that
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is the statement:
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| − |
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| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
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| − | i
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| − | +
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| − | n
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| − | {\displaystyle i+n} vertices, then
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}} for some
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| − | i
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| − | <
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| − | j
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| − | {\displaystyle i<j}.
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| − | All the statements
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
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| − | P
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| − | (
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| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
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| − |
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| − | Define
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| − | tree
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
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| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
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| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
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| − | It is known that
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| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
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| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
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| − | tree
| |
| − | (
| |
| − | 2
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| − | )
| |
| − | =
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| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
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| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
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| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
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| − | 131
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| − | ,
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| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
| |
| − | vte
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| − | Large numbers
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| − | vte
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| − | Order theory
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| − | Categories: Mathematical logicOrder theoryTheorems in discrete mathematicsTrees (graph theory)Wellfoundedness
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| − | This page was last edited on 21 October 2024, at 22:33 (UTC).
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
| |
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | TREE function
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| − | Kruskal's tree theorem
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
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| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | TREE function
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| − | Kruskal's tree theorem
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
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| − |
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| − | History
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| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
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| − |
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| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
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| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
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| − |
| |
| − | Statement
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| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
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| − |
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| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
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| − |
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| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
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| − | T
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| − | 1
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| − | ≤
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| − | T
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| − | 2
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| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
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| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
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| − | F
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| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
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| − | If w is any successor of v in T1, then
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| − | F
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| − | (
| |
| − | w
| |
| − | )
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| − | {\displaystyle F(w)} is a successor of
| |
| − | F
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| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
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| − | If w1, w2 are any two distinct immediate successors of v, then the path from
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| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
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| − | F
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| − | (
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| − | w
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| − | 2
| |
| − | )
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| − | {\displaystyle F(w_{2})} in T2 contains
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| − | F
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| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
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| − | Kruskal's tree theorem then states:
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| − |
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| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
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| − | {\displaystyle T_{i}\leq T_{j}}.)
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| − |
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| − | Friedman's work
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| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
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| − |
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| − | Weak tree function
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| − | Suppose that
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| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
| |
| − | vte
| |
| − | Large numbers
| |
| − | vte
| |
| − | Order theory
| |
| − | Categories: Mathematical logicOrder theoryTheorems in discrete mathematicsTrees (graph theory)Wellfoundedness
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| − | Friedman's work
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| − | Weak tree function
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| − | TREE function
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| − | Kruskal's tree theorem
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
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| − |
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| − | History
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| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
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| − |
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| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
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| − | TREE
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| − | (
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| − | 3
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| − | )
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| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
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| − |
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| − | Statement
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| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
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| − |
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| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
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| − |
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| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
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| − | T
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| − | 1
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| − | ≤
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| − | T
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| − | 2
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| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
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| − |
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| − | For all vertices v of T1, the label of v precedes the label of
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)};
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| − | If w is any successor of v in T1, then
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| − | F
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| − | (
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| − | w
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| − | )
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| − | {\displaystyle F(w)} is a successor of
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)}; and
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| − | If w1, w2 are any two distinct immediate successors of v, then the path from
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| − | F
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| − | (
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| − | w
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| − | 1
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| − | )
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| − | {\displaystyle F(w_{1})} to
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| − | F
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| − | (
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| − | w
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| − | 2
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| − | )
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| − | {\displaystyle F(w_{2})} in T2 contains
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)}.
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| − | Kruskal's tree theorem then states:
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| − |
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| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
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| − | i
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| − | <
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| − | j
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| − | {\displaystyle i<j} so that
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}}.)
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| − |
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| − | Friedman's work
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| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
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| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
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| − | 1-CA0.
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| − |
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| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
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| − |
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| − | Weak tree function
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| − | Suppose that
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is the statement:
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| − |
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| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
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| − | i
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| − | +
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| − | n
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| − | {\displaystyle i+n} vertices, then
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}} for some
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| − | i
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| − | <
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| − | j
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| − | {\displaystyle i<j}.
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| − | All the statements
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
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| − |
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| − | Define
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| − | tree
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
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| − |
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| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
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| − | i
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| − | +
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| − | n
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| − | {\displaystyle i+n} vertices, such that
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
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| − | i
| |
| − | <
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| − | j
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| − | ≤
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| − | m
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| − | {\displaystyle i<j\leq m}.
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| − | It is known that
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| − | tree
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| − | (
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| − | 1
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| − | )
| |
| − | =
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| − | 2
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| − | {\displaystyle {\text{tree}}(1)=2},
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| − | tree
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| − | (
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| − | 2
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| − | )
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| − | =
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| − | 5
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| − | {\displaystyle {\text{tree}}(2)=5},
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| − | tree
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| − | (
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| − | 3
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| − | )
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| − | ≥
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| − | 844
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| − | ,
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| − | 424
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| − | ,
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| − | 930
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| − | ,
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| − | 131
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| − | ,
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| − | 960
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| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
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| − | tree
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| − | (
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| − | 4
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| − | )
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| − | ≫
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| − | g
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| − | 64
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| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
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| − | g
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| − | 64
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| − | {\displaystyle g_{64}} is Graham's number), and
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| − | TREE
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| − | (
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| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
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| − | t
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| − | r
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| − | e
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| − | e
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| − | t
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| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
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| − | e
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| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
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| − | 7
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| − | )
| |
| − | .
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| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
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| − |
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| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
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| − |
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| − | TREE function
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| − | Sequence of trees where each node is colored either green, red, blue
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| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
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| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
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| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
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| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
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| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
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| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
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| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
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| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | Categories: Mathematical logicOrder theoryTheorems in discrete mathematicsTrees (graph theory)Wellfoundedness
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| − | This page was last edited on 21 October 2024, at 22:33 (UTC).
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
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| − |
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| − | History
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| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
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| − |
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| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
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| − | TREE
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| − | (
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| − | 3
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| − | )
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| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
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| − |
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| − | Statement
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| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
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| − |
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| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
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| − |
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| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
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| − | T
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| − | 1
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| − | ≤
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| − | T
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| − | 2
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| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
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| − |
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| − | For all vertices v of T1, the label of v precedes the label of
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)};
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| − | If w is any successor of v in T1, then
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| − | F
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| − | (
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| − | w
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| − | )
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| − | {\displaystyle F(w)} is a successor of
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)}; and
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| − | If w1, w2 are any two distinct immediate successors of v, then the path from
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| − | F
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| − | (
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| − | w
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| − | 1
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| − | )
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| − | {\displaystyle F(w_{1})} to
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| − | F
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| − | (
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| − | w
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| − | 2
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| − | )
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| − | {\displaystyle F(w_{2})} in T2 contains
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)}.
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| − | Kruskal's tree theorem then states:
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| − |
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| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
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| − | i
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| − | <
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| − | j
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| − | {\displaystyle i<j} so that
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}}.)
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| − |
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| − | Friedman's work
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| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
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| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
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| − | 1-CA0.
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| − |
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| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
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| − |
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| − | Weak tree function
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| − | Suppose that
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is the statement:
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| − |
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| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
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| − | i
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| − | +
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| − | n
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| − | {\displaystyle i+n} vertices, then
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}} for some
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| − | i
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| − | <
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| − | j
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| − | {\displaystyle i<j}.
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| − | All the statements
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
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| − |
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| − | Define
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| − | tree
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
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| − |
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| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
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| − | i
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| − | +
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| − | n
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| − | {\displaystyle i+n} vertices, such that
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
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| − | i
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| − | <
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| − | j
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| − | ≤
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| − | m
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| − | {\displaystyle i<j\leq m}.
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| − | It is known that
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| − | tree
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| − | (
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| − | 1
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| − | )
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| − | =
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| − | 2
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| − | {\displaystyle {\text{tree}}(1)=2},
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| − | tree
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| − | (
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| − | 2
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| − | )
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| − | =
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| − | 5
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| − | {\displaystyle {\text{tree}}(2)=5},
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| − | tree
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| − | (
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| − | 3
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| − | )
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| − | ≥
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| − | 844
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| − | ,
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| − | 424
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| − | ,
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| − | 930
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| − | ,
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| − | 131
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| − | ,
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| − | 960
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| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
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| − | tree
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| − | (
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| − | 4
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| − | )
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| − | ≫
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| − | g
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| − | 64
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| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
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| − | g
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| − | 64
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| − | {\displaystyle g_{64}} is Graham's number), and
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| − | TREE
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| − | (
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| − | 3
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| − | )
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| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
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| − | t
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| − | r
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| − | e
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| − | e
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| − | t
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| − | r
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| − | e
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| − | e
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| − | t
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| − | r
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| − | e
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| − | e
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| − | t
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| − | r
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| − | e
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| − | e
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| − | t
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| − | r
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| − | e
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| − | e
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| − | 8
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| − | (
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| − | 7
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| − | )
| |
| − | (
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| − | 7
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| − | )
| |
| − | (
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| − | 7
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| − | )
| |
| − | (
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| − | 7
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| − | )
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| − | (
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| − | 7
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| − | )
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| − | .
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| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
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| − |
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| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
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| − |
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| − | TREE function
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| − | Sequence of trees where each node is colored either green, red, blue
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| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
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| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
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| − | TREE
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
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| − |
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| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
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| − | i
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| − | <
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| − | j
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| − | ≤
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| − | m
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| − | {\displaystyle i<j\leq m}.
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| − | The TREE sequence begins
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| − | TREE
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| − | (
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| − | 1
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| − | )
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| − | =
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| − | 1
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| − | {\displaystyle {\text{TREE}}(1)=1},
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| − | TREE
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| − | (
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| − | 2
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| − | )
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| − | =
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| − | 3
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| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
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| − | TREE
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| − | (
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| − | 3
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| − | )
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| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
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| − | n
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| − | (
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| − | 4
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| − | )
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| − | {\displaystyle n(4)},
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| − | n
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| − | n
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| − | (
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| − | 5
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| − | )
| |
| − | (
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| − | 5
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| − | )
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| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
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| − | n
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| − | (
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| − | 4
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| − | )
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| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
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| − | TREE
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| − | (
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| − | 3
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| − | )
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| − | {\displaystyle {\text{TREE}}(3)}, is
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| − | A
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| − | A
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| − | (
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| − | 187196
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| − | )
| |
| − | (
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| − | 1
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| − | )
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| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
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| − | A
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| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
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| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
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| − | g
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| − | 3
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| − | ↑
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| − | 187196
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| − | 3
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| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
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| − | g
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| − | x
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| − | {\displaystyle g_{x}} is Graham's function.
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| − |
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| − | See also
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| − | Paris–Harrington theorem
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| − | Kanamori–McAloon theorem
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| − | Robertson–Seymour theorem
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| − | Notes
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| − | ^ a Friedman originally denoted this function by TR[n].
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| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
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| − | 11
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| − | ,
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| − | and
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| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
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| − | 2
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| − | ↑
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| − | 7197
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| − | 158386
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| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
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| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
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| − | References
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| − | Citations
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| − |
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| − | Simpson 1985, Theorem 1.8
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| − | Friedman 2002, p. 60
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| − | Simpson 1985, Definition 4.1
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| − | Simpson 1985, Theorem 5.14
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| − | Marcone 2001, pp. 8–9
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| − | Rathjen & Weiermann 1993.
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| − | Smith 1985, p. 120
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| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
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| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
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| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
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| − | Bibliography
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| − |
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| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
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| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
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| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
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| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
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| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
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| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
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| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | Large numbers
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| − | vte
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| − | Order theory
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| − | Categories: Mathematical logicOrder theoryTheorems in discrete mathematicsTrees (graph theory)Wellfoundedness
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| − | This page was last edited on 21 October 2024, at 22:33 (UTC).
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
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| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
| |
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
| |
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| − | TREE function
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| − | Notes
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| − | References
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| − | Kruskal's tree theorem
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | Kruskal's tree theorem
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
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| − |
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| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
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| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
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| − |
| |
| − | Statement
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| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
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| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
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| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
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| − | 1
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| − | ≤
| |
| − | T
| |
| − | 2
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| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
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| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
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| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
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| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
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| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
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| − | Kruskal's tree theorem then states:
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| − |
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| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
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| − |
| |
| − | Friedman's work
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| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | Categories: Mathematical logicOrder theoryTheorems in discrete mathematicsTrees (graph theory)Wellfoundedness
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| − | TREE function
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| − | Kruskal's tree theorem
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
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| − |
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| − | History
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| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
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| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
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| − |
| |
| − | Statement
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| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
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| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
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| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
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| − | 1
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| − | ≤
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| − | T
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| − | 2
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| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
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| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
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| − | F
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| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
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| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
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| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
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| − | (
| |
| − | w
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| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
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| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
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| − | Kruskal's tree theorem then states:
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| − |
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| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
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| − | {\displaystyle T_{i}\leq T_{j}}.)
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| − |
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| − | Friedman's work
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| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
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| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
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| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
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| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | Order theory
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| − | Categories: Mathematical logicOrder theoryTheorems in discrete mathematicsTrees (graph theory)Wellfoundedness
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| − | This page was last edited on 21 October 2024, at 22:33 (UTC).
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| − | From Wikipedia, the free encyclopedia
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| − |
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
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| − |
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| − | History
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| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
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| − |
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| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
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| − | TREE
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| − | (
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| − | 3
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| − | )
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| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
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| − |
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| − | Statement
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| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
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| − |
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| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
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| − |
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| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
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| − | T
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| − | 1
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| − | ≤
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| − | T
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| − | 2
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| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
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| − |
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| − | For all vertices v of T1, the label of v precedes the label of
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| − | F
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| − | (
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| − | v
| |
| − | )
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| − | {\displaystyle F(v)};
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| − | If w is any successor of v in T1, then
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| − | F
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| − | (
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| − | w
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| − | )
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| − | {\displaystyle F(w)} is a successor of
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)}; and
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| − | If w1, w2 are any two distinct immediate successors of v, then the path from
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| − | F
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| − | (
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| − | w
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| − | 1
| |
| − | )
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| − | {\displaystyle F(w_{1})} to
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| − | F
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| − | (
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| − | w
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| − | 2
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| − | )
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| − | {\displaystyle F(w_{2})} in T2 contains
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| − | F
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| − | (
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| − | v
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| − | )
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| − | {\displaystyle F(v)}.
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| − | Kruskal's tree theorem then states:
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| − |
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| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
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| − | j
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| − | {\displaystyle i<j} so that
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}}.)
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| − |
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| − | Friedman's work
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| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
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| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
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| − | 1-CA0.
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| − |
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| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
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| − |
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| − | Weak tree function
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| − | Suppose that
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| − | P
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| − | (
| |
| − | n
| |
| − | )
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| − | {\displaystyle P(n)} is the statement:
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| − |
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| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
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| − | i
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| − | +
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| − | n
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| − | {\displaystyle i+n} vertices, then
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| − | T
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| − | i
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| − | ≤
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| − | T
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| − | j
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| − | {\displaystyle T_{i}\leq T_{j}} for some
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| − | i
| |
| − | <
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| − | j
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| − | {\displaystyle i<j}.
| |
| − | All the statements
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| − | P
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| − | (
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| − | n
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| − | )
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| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
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| − | P
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| − | (
| |
| − | n
| |
| − | )
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| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
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| − | (
| |
| − | n
| |
| − | )
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| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
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| − | P
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| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
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| − | P
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| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
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| − |
| |
| − | Define
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| − | tree
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| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
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| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
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| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
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| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
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| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
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| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | Categories: Mathematical logicOrder theoryTheorems in discrete mathematicsTrees (graph theory)Wellfoundedness
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
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| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
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| − | From Wikipedia, the free encyclopedia
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| − | This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2024) (Learn how and when to remove this message)
| |
| − | In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
| |
| − |
| |
| − | History
| |
| − | The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963). It has since become a prominent example in reverse mathematics as a statement that cannot be proved in ATR0 (a second-order arithmetic theory with a form of arithmetical transfinite recursion).
| |
| − |
| |
| − | In 2004, the result was generalized from trees to graphs as the Robertson–Seymour theorem, a result that has also proved important in reverse mathematics and leads to the even-faster-growing SSCG function, which dwarfs
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}. A finitary application of the theorem gives the existence of the fast-growing TREE function.
| |
| − |
| |
| − | Statement
| |
| − | The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite.
| |
| − |
| |
| − | Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
| |
| − |
| |
| − | Take X to be a partially ordered set. If T1, T2 are rooted trees with vertices labeled in X, we say that T1 is inf-embeddable in T2 and write
| |
| − | T
| |
| − | 1
| |
| − | ≤
| |
| − | T
| |
| − | 2
| |
| − | {\displaystyle T_{1}\leq T_{2}} if there is an injective map F from the vertices of T1 to the vertices of T2 such that:
| |
| − |
| |
| − | For all vertices v of T1, the label of v precedes the label of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)};
| |
| − | If w is any successor of v in T1, then
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | )
| |
| − | {\displaystyle F(w)} is a successor of
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}; and
| |
| − | If w1, w2 are any two distinct immediate successors of v, then the path from
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle F(w_{1})} to
| |
| − | F
| |
| − | (
| |
| − | w
| |
| − | 2
| |
| − | )
| |
| − | {\displaystyle F(w_{2})} in T2 contains
| |
| − | F
| |
| − | (
| |
| − | v
| |
| − | )
| |
| − | {\displaystyle F(v)}.
| |
| − | Kruskal's tree theorem then states:
| |
| − |
| |
| − | If X is well-quasi-ordered, then the set of rooted trees with labels in X is well-quasi-ordered under the inf-embeddable order defined above. (That is to say, given any infinite sequence T1, T2, … of rooted trees labeled in X, there is some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j} so that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}}.)
| |
| − |
| |
| − | Friedman's work
| |
| − | For a countable label set X, Kruskal's tree theorem can be expressed and proven using second-order arithmetic. However, like Goodstein's theorem or the Paris–Harrington theorem, some special cases and variants of the theorem can be expressed in subsystems of second-order arithmetic much weaker than the subsystems where they can be proved. This was first observed by Harvey Friedman in the early 1980s, an early success of the then-nascent field of reverse mathematics. In the case where the trees above are taken to be unlabeled (that is, in the case where X has size one), Friedman found that the result was unprovable in ATR0,[1] thus giving the first example of a predicative result with a provably impredicative proof.[2] This case of the theorem is still provable by Π1
| |
| − | 1-CA0, but by adding a "gap condition"[3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system.[4][5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π1
| |
| − | 1-CA0.
| |
| − |
| |
| − | Ordinal analysis confirms the strength of Kruskal's theorem, with the proof-theoretic ordinal of the theorem equaling the small Veblen ordinal (sometimes confused with the smaller Ackermann ordinal).[6]
| |
| − |
| |
| − | Weak tree function
| |
| − | Suppose that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is the statement:
| |
| − |
| |
| − | There is some m such that if T1, ..., Tm is a finite sequence of unlabeled rooted trees where Ti has
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, then
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} for some
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | {\displaystyle i<j}.
| |
| − | All the statements
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} are true as a consequence of Kruskal's theorem and Kőnig's lemma. For each n, Peano arithmetic can prove that
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true, but Peano arithmetic cannot prove the statement "
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} is true for all n".[7] Moreover, the length of the shortest proof of
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} in Peano arithmetic grows phenomenally fast as a function of n, far faster than any primitive recursive function or the Ackermann function, for example.[citation needed] The least m for which
| |
| − | P
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle P(n)} holds similarly grows extremely quickly with n.
| |
| − |
| |
| − | Define
| |
| − | tree
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{tree}}(n)}, the weak tree function, as the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of unlabeled rooted trees, where each Ti has at most
| |
| − | i
| |
| − | +
| |
| − | n
| |
| − | {\displaystyle i+n} vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | It is known that
| |
| − | tree
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 2
| |
| − | {\displaystyle {\text{tree}}(1)=2},
| |
| − | tree
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 5
| |
| − | {\displaystyle {\text{tree}}(2)=5},
| |
| − | tree
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | ≥
| |
| − | 844
| |
| − | ,
| |
| − | 424
| |
| − | ,
| |
| − | 930
| |
| − | ,
| |
| − | 131
| |
| − | ,
| |
| − | 960
| |
| − | {\displaystyle {\text{tree}}(3)\geq 844,424,930,131,960} (about 844 trillion),
| |
| − | tree
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | ≫
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle {\text{tree}}(4)\gg g_{64}} (where
| |
| − | g
| |
| − | 64
| |
| − | {\displaystyle g_{64}} is Graham's number), and
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} (where the argument specifies the number of labels; see below) is larger than
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | t
| |
| − | r
| |
| − | e
| |
| − | e
| |
| − | 8
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | (
| |
| − | 7
| |
| − | )
| |
| − | .
| |
| − | {\displaystyle \mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{\mathrm {tree} ^{8}(7)}(7)}(7)}(7)}(7).}
| |
| − |
| |
| − | To differentiate the two functions, "TREE" (with all caps) is the big TREE function, and "tree" (with all letters in lowercase) is the weak tree function.
| |
| − |
| |
| − | TREE function
| |
| − | Sequence of trees where each node is colored either green, red, blue
| |
| − | A sequence of rooted trees labelled from a set of 3 labels (blue < red < green). The nth tree in the sequence contains at most n vertices, and no tree is inf-embeddable within any later tree in the sequence. TREE(3) is defined to be the longest possible length of such a sequence.
| |
| − | By incorporating labels, Friedman defined a far faster-growing function.[8] For a positive integer n, take
| |
| − | TREE
| |
| − | (
| |
| − | n
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(n)}[a] to be the largest m so that we have the following:
| |
| − |
| |
| − | There is a sequence T1, ..., Tm of rooted trees labelled from a set of n labels, where each Ti has at most i vertices, such that
| |
| − | T
| |
| − | i
| |
| − | ≤
| |
| − | T
| |
| − | j
| |
| − | {\displaystyle T_{i}\leq T_{j}} does not hold for any
| |
| − | i
| |
| − | <
| |
| − | j
| |
| − | ≤
| |
| − | m
| |
| − | {\displaystyle i<j\leq m}.
| |
| − | The TREE sequence begins
| |
| − | TREE
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 1
| |
| − | {\displaystyle {\text{TREE}}(1)=1},
| |
| − | TREE
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | {\displaystyle {\text{TREE}}(2)=3}, then suddenly,
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)} explodes to a value that is so big that many other "large" combinatorial constants, such as Friedman's
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)},
| |
| − | n
| |
| − | n
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | (
| |
| − | 5
| |
| − | )
| |
| − | {\displaystyle n^{n(5)}(5)}, and Graham's number,[b] are extremely small by comparison. A lower bound for
| |
| − | n
| |
| − | (
| |
| − | 4
| |
| − | )
| |
| − | {\displaystyle n(4)}, and, hence, an extremely weak lower bound for
| |
| − | TREE
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | {\displaystyle {\text{TREE}}(3)}, is
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}.[c][9] Graham's number, for example, is much smaller than the lower bound
| |
| − | A
| |
| − | A
| |
| − | (
| |
| − | 187196
| |
| − | )
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | {\displaystyle A^{A(187196)}(1)}, which is approximately
| |
| − | g
| |
| − | 3
| |
| − | ↑
| |
| − | 187196
| |
| − | 3
| |
| − | {\displaystyle g_{3\uparrow ^{187196}3}}, where
| |
| − | g
| |
| − | x
| |
| − | {\displaystyle g_{x}} is Graham's function.
| |
| − |
| |
| − | See also
| |
| − | Paris–Harrington theorem
| |
| − | Kanamori–McAloon theorem
| |
| − | Robertson–Seymour theorem
| |
| − | Notes
| |
| − | ^ a Friedman originally denoted this function by TR[n].
| |
| − | ^ b n(k) is defined as the length of the longest possible sequence that can be constructed with a k-letter alphabet such that no block of letters xi,...,x2i is a subsequence of any later block xj,...,x2j.[10]
| |
| − | n
| |
| − | (
| |
| − | 1
| |
| − | )
| |
| − | =
| |
| − | 3
| |
| − | ,
| |
| − | n
| |
| − | (
| |
| − | 2
| |
| − | )
| |
| − | =
| |
| − | 11
| |
| − | ,
| |
| − | and
| |
| − | n
| |
| − | (
| |
| − | 3
| |
| − | )
| |
| − | >
| |
| − | 2
| |
| − | ↑
| |
| − | 7197
| |
| − | 158386
| |
| − | {\displaystyle n(1)=3,n(2)=11,\,{\textrm {and}}\,n(3)>2\uparrow ^{7197}158386}.
| |
| − | ^ c A(x) taking one argument, is defined as A(x, x), where A(k, n), taking two arguments, is a particular version of Ackermann's function defined as: A(1, n) = 2n, A(k+1, 1) = A(k, 1), A(k+1, n+1) = A(k, A(k+1, n)).
| |
| − | References
| |
| − | Citations
| |
| − |
| |
| − | Simpson 1985, Theorem 1.8
| |
| − | Friedman 2002, p. 60
| |
| − | Simpson 1985, Definition 4.1
| |
| − | Simpson 1985, Theorem 5.14
| |
| − | Marcone 2001, pp. 8–9
| |
| − | Rathjen & Weiermann 1993.
| |
| − | Smith 1985, p. 120
| |
| − | Friedman, Harvey (28 March 2006). "273:Sigma01/optimal/size". Ohio State University Department of Maths. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (1 June 2000). "Enormous Integers In Real Life" (PDF). Ohio State University. Retrieved 8 August 2017.
| |
| − | Friedman, Harvey M. (8 October 1998). "Long Finite Sequences" (PDF). Ohio State University Department of Mathematics. pp. 5, 48 (Thm.6.8). Retrieved 8 August 2017.
| |
| − | Bibliography
| |
| − |
| |
| − | Friedman, Harvey M. (2002). "Internal finite tree embeddings". In Sieg, Wilfried; Feferman, Solomon (eds.). Reflections on the foundations of mathematics: essays in honor of Solomon Feferman. Lecture notes in logic. Vol. 15. Natick, Mass: AK Peters. pp. 60–91. ISBN 978-1-56881-170-3. MR 1943303.
| |
| − | H. Gallier, Jean (September 1991). "What's so special about Kruskal's theorem and the ordinal Γ0? A survey of some results in proof theory" (PDF). Annals of Pure and Applied Logic. 53 (3): 199–260. doi:10.1016/0168-0072(91)90022-E. MR 1129778.
| |
| − | Kruskal, J. B. (May 1960). "Well-Quasi-Ordering, The Tree Theorem, and Vazsonyi's Conjecture" (PDF). Transactions of the American Mathematical Society. 95 (2). American Mathematical Society: 210–225. doi:10.2307/1993287. JSTOR 1993287. MR 0111704.
| |
| − | Marcone, Alberto (2005). Simpson, Stephen G. (ed.). "WQO and BQO theory in subsystems of second order arithmetic" (PDF). Reverse Mathematics. Lecture Notes in Logic. 21. Cambridge: Cambridge University Press: 303–330. doi:10.1017/9781316755846.020. ISBN 978-1-316-75584-6.
| |
| − | Nash-Williams, C. St. J. A. (October 1963). "On well-quasi-ordering finite trees" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 59 (4): 833–835. Bibcode:1963PCPS...59..833N. doi:10.1017/S0305004100003844. ISSN 0305-0041. MR 0153601. S2CID 251095188.
| |
| − | Rathjen, Michael; Weiermann, Andreas (February 1993). "Proof-theoretic investigations on Kruskal's theorem" (PDF). Annals of Pure and Applied Logic. 60 (1): 49–88. doi:10.1016/0168-0072(93)90192-G. MR 1212407.
| |
| − | Simpson, Stephen G. (1985). "Nonprovability of certain combinatorial properties of finite trees". In Friedman, Harvey; Harrington, L. A.; Scedrov, A.; et al. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Amsterdam ; New York: North-Holland. pp. 87–117. ISBN 978-0-444-87834-2.
| |
| − | Smith, Rick L. (1985). "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems". In Friedman, Harvey; Harrington, L. A. (eds.). Harvey Friedman's research on the foundations of mathematics. Studies in logic and the foundations of mathematics. Vol. 117. Amsterdam ; New York: North-Holland. pp. 119–136. doi:10.1016/s0049-237x(09)70157-0. ISBN 978-0-444-87834-2.
| |
| − | vte
| |
| − | Large numbers
| |
| − | vte
| |
| − | Order theory
| |
| − | Categories: Mathematical logicOrder theoryTheorems in discrete mathematicsTrees (graph theory)Wellfoundedness
| |
| − | This page was last edited on 21 October 2024, at 22:33 (UTC).
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