|(mathematics)||von Neumann ordinal - An implementation of ordinals in set theory
(e.g. Zermelo Fränkel set theory or ZFC). The von Neumann
ordinal alpha is the well-ordered set containing just the
ordinals "shorter" than alpha.|
"Reasonable" set theories (like ZF) include Mostowski's Collapsing Theorem: any well-ordered set is isomorphic to a von Neumann ordinal. In really screwy theories (e.g. NFU -- New Foundations with Urelemente) this theorem is false.
The finite von Neumann ordinals are the von Neumann integers.