von Neumann ordinal

An implementation of {ordinal}s in {set theory}

(e.g. {Zermelo Fr�nkel 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

integer}s.

(1995-03-30)