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The von Neumann ordinal is a method of defining ordinals in set theory.
The von Neumann ordinal  is defined to be the well-ordered set containing the von Neumann ordinals which precede  . 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
 , the ordinal  (the successor of  is defined to be
 ).
- Given a set
 of ordinals,
 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  .
The class of ordinals is denoted
 .
The von Neumann ordinals have the convenient property that if  then  and
 .
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