# von Neumann ordinal

The is a method of defining ordinals in set theory.

The von Neumann ordinal $\alpha$ is defined to be the well-ordered set containing the von Neumann ordinals which precede $\alpha$. 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:

• The empty set is $0$.

• Given any ordinal $\alpha$, the ordinal $\alpha+1$ (the successor of $\alpha$) is defined to be $\alpha\cup\{\alpha\}$.

• Given a set $A$ of ordinals, $\bigcup_{a\in A}a$ is an ordinal.

If an ordinal is the successor of another ordinal, it is an successor ordinal. If an ordinal is neither $0$ nor a successor ordinal then it is a limit ordinal. The first limit ordinal is named $\omega$.

The class of ordinals is denoted $\mathbf{On}$.

The von Neumann ordinals have the convenient property that if $a then $a\in b$ and $a\subset b$.

 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