Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

User:Evin-/Draft:Ordinal

Revision as of 11:23, 5 June 2020 by Evin- (talk | contribs) (→‎The Veblen \phi functions)

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, $\omega$. Ordinals can be added and multiplied. The sum of two ordinals $a$ and $b$ 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 $1+\omega=\omega$, while $\omega+1>\omega$.

Every ordinal characterizes the order type of the ordered ordinals less than it. For example, $0,1,2,\dotsb,\omega$ has order type $\omega+1$.

The smallest ordinal that can't be constructed from $\omega$ by addition, multiplication, and exponentiation is $\varepsilon_0$, the first fixed point of the map $\alpha\mapsto\omega^\alpha$.

The Veblen $\phi$ functions

Based on the definition of $\varepsilon_0$, Oswald Veblen in 1908 introduced an ordinal-indexed hierarchy of functions. $\phi_0(n)=\omega^n$ and $\phi_k$ enumerates the common fixed points of $\phi_m$ for all $m<k$. $\phi_1(0)=\varepsilon_0$, $\phi_2(0)$ is the smallest ordinal inaccessible from the $\varepsilon$ ordinals. It is called $\zeta_0$. The set of all ordinals accessible from the $\phi$ functions, addition, multiplication, and exponentiation is well-ordered. It has order type $\Gamma_0$, the Feferman–Schütte ordinal. It is the first fixed point of the map $\alpha\mapsto\phi_\alpha(0)$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=User:Evin-/Draft:Ordinal&oldid=123876"