 The axiom of infinity is usually now modified to assert the existence of the first infinite
von Neumann ordinal formula_2; the original Zermelo
axioms cannot prove the existence of this set, nor can the modified Zermelo axioms prove Zermelo's
axiom of infinity.Zermelo's axioms (original or modified) cannot prove the existence of formula_3 as a set nor of any rank of the cumulative hierarchy of sets with infinite index.Zermelo allowed for the existence of urelements that are not sets and contain no elements; these are now usually omitted from set theories.