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 $\mathrm{𝐎𝐧}$.

The von Neumann ordinals have the convenient property that if 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