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 regularity of a habit is generally in proportion to its absurdity.”
—Marcel Proust (18711922)
“When the weather is bad as it was yesterday, everybody, almost everybody, feels cross and gloomy. Our thin linen tentsabout like a fish seine, the deep mud, the irregular mails, the never to-be-seen paymasters, and the rest of mankind, are growled about in old-soldier style. But a fine day like today has turned out brightens and cheers us all. We people in camp are merely big children, wayward and changeable.”
—Rutherford Birchard Hayes (18221893)
“The axioms of physics translate the laws of ethics. Thus, the whole is greater than its part; reaction is equal to action; the smallest weight may be made to lift the greatest, the difference of weight being compensated by time; and many the like propositions, which have an ethical as well as physical sense. These propositions have a much more extensive and universal sense when applied to human life, than when confined to technical use.”
—Ralph Waldo Emerson (18031882)