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)