Difference between revisions of "User:Evin-/Draft:Ordinal"
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The smallest ordinal that can't be constructed from <math>\omega</math> by addition, multiplication, and exponentiation is <math>\varepsilon_0</math>, the first fixed point of the map <math>\alpha\mapsto\omega^\alpha</math>. | The smallest ordinal that can't be constructed from <math>\omega</math> by addition, multiplication, and exponentiation is <math>\varepsilon_0</math>, the first fixed point of the map <math>\alpha\mapsto\omega^\alpha</math>. | ||
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| + | == The Veblen <math>\phi</math> functions == | ||
| + | Based on the definition of <math>\varepsilon_0</math>, Oswald Veblen in 1908 introduced an ordinal-indexed hierarchy of functions. <math>\phi_0(n)=\omega^n</math> and <math>\phi_k</math> enumerates the common fixed points of <math>\phi_m</math> for all <math>m<k</math>. | ||
Revision as of 10:13, 5 June 2020
Ordinals are an extension of the natural numbers. Ordinals can be used to describe the order type of a set. The order type of the natural numbers is the first infinite ordinal, 
. Ordinals can be added and multiplied. The sum of two ordinals 
and 
is the ordinal that describes the order type of a set with order type a concatenated with one of order type b. Warning! Ordinal addition is not commutative. For example 
, while 
.
Every ordinal characterizes the order type of the ordered ordinals less than it. For example, 
has order type 
.
The smallest ordinal that can't be constructed from by addition, multiplication, and exponentiation is 
, the first fixed point of the map 
.
The Veblen 
functions
functionsBased on the definition of , Oswald Veblen in 1908 introduced an ordinal-indexed hierarchy of functions. 
and 
enumerates the common fixed points of 
for all 
.
