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)