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