von Neumann ordinal
<mathematics> 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}.
(1995-03-30)