von Neumann ordinal
The is a method of defining ordinals in set theory.
The von Neumann ordinal is defined to be the well-ordered set containing the von Neumann ordinals which precede . The set of finite von Neumann ordinals is known as the von Neumann integers. Every well-ordered set is isomorphic to a von Neumann ordinal.
They can be constructed by transfinite recursion as follows:
If an ordinal is the successor of another ordinal, it is an successor ordinal. If an ordinal is neither nor a successor ordinal then it is a limit ordinal. The first limit ordinal is named .
The class of ordinals is denoted .
The von Neumann ordinals have the convenient property that if then and .
Title | von Neumann ordinal |
Canonical name | VonNeumannOrdinal |
Date of creation | 2013-03-22 12:32:37 |
Last modified on | 2013-03-22 12:32:37 |
Owner | Henry (455) |
Last modified by | Henry (455) |
Numerical id | 11 |
Author | Henry (455) |
Entry type | Definition |
Classification | msc 03E10 |
Synonym | ordinal |
Related topic | VonNeumannInteger |
Related topic | ZermeloFraenkelAxioms |
Related topic | OrdinalNumber |
Defines | successor ordinal |
Defines | limit ordinal |
Defines | successor |