Axiom of Regularity - Regularity and The Rest of ZF(C) Axioms

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)

    While the rest of the world has been improving technology, Ghana has been improving the quality of man’s humanity to man.
    Maya Angelou (b. 1928)

    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 (1803–1882)