Source : Free On-Line Dictionary of Computing

von Neumann ordinal
     
         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 Neumann
        integers}.
     
        (1995-03-30)