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From The Free On-line Dictionary of Computing (05 January 2017) [foldoc]:
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)