The Free On-line Dictionary of Computing (30 December 2018):

von Neumann ordinal

    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)