Editing Axiom of regularity (section)

== Regularity and the rest of ZF(C) axioms ==
Regularity was shown to be relatively consistent with the rest of ZF by Skolem{{sfn|Skolem|1923}} and von Neumann,{{sfn|von Neumann|1929}} meaning that if ZF without regularity is consistent, then ZF (with regularity) is also consistent.<ref>For his{{ambiguous|reason=Skolem or Von Neumann?|date=December 2024}} proof in modern notation, see {{harvtxt|Vaught|2001|loc=§10.1}} for instance.</ref>

The axiom of regularity was also shown to be [[Independence (mathematical logic)|independent]] from the other axioms of ZFC, assuming they are consistent. The result was announced by [[Paul Bernays]] in 1941, although he did not publish a proof until 1954. The proof involves (and led to the study of) Rieger-Bernays [[permutation model]]s (or method), which were used for other proofs of independence for non-well-founded systems.{{sfn|Rathjen|2004|p=193}}{{sfn|Forster|2003|pp=210–212}}