## You are here

Homevon Neumann ordinal

## Primary tabs

# von Neumann ordinal

The *von Neumann ordinal* 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:

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 $a<b$ then $a\in b$ and $a\subset b$.

## Mathematics Subject Classification

03E10*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

## Attached Articles

## Corrections

missing word by Wkbj79 ✓

missing ) by yark ✓

Placement of ) in second item by MikeFikes ✓