Regularity and The Rest of ZF(C) Axioms
Regularity was shown to be relatively consistent with the rest of ZF by von Neumann (1929), meaning that if ZF without regularity is consistent, then ZF (with regularity) is also consistent. For his proof in modern notation see Vaught (2001, §10.1) for instance.
The axiom of regularity was also shown to be independent from the other axioms of ZF(C), 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 models (or method), which were used for other proofs of independence for non-well-founded systems (Rathjen 2004, p. 193 and Forster 2003, pp. 210–212).
Read more about this topic: Axiom Of Regularity
Famous quotes containing the words regularity, rest and/or axioms:
“Generality is, indeed, an indispensable ingredient of reality; for mere individual existence or actuality without any regularity whatever is a nullity. Chaos is pure nothing.”
—Charles Sanders Peirce (1839–1914)
“The miracles of the church seem to me to rest not so much upon faces or voices or healing power coming suddenly near to us from afar off, but upon our perceptions being made finer, so that for a moment our eyes can see and our ears can hear what is there about us always.”
—Willa Cather (1873–1947)
““I tell you the solemn truth that the doctrine of the Trinity is not so difficult to accept for a working proposition as any one of the axioms of physics.””
—Henry Brooks Adams (1838–1918)