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von neumann ordinal : FOLDOC | von Neumann ordinal An implementation of {ordinals} in set theory (e.g. Zermelo Frnkel 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)
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