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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). Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Privacy policyAbout WikipediaDisclaimersContact WikipediaCode of ConductDevelopersStatisticsCookie statementMobile view Wikimedia FoundationPowered by MediaWiki