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:
“The man of business ... goes on Sunday to the church with the regularity of the village blacksmith, there to renounce and abjure before his God the line of conduct which he intends to pursue with all his might during the following week.”
—George Bernard Shaw (1856–1950)
“Thinking about the universe has now been handed over to specialists. The rest of us merely read about it.”
—Mason Cooley (b. 1927)
““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)