von Neumann ordinal

mathematics An implementation of ordinals in set theory

(e.g. Zermelo Frankel 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 Neumannintegers.

(1995-03-30)