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# 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*

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