On the consistency of 'New Foundations'. Offprint from: Proceedings of the National Academy of Sciences, vol. 37, no. 8, August, 1951

QUINE, Willard van Orman

Erscheinungsdatum: 1951

Beschreibung

First edition, the rare offprint, in which Quine proves that "NF is consistent if and only if the system of Whitehead and Russell's Principia Mathematica, briefly PM, is consistent in a certain one of its natural versions' (p. 538). "Although [Quine] is best known to a wider public for his philosophical writings, his most enduring and most concrete legacy for the next 50 years may well turn out to be his most mathematical: he gave us NF" (Forster, p. 838). NF was intended to address the 'crisis of foundations' that mathematicians have attempted to resolve since the early 20th century. "This 'crisis' had many causes and - despite the disappearance of the expression from contemporary speech - has never really been resolved. One of its many causes was the increasing formalisation of mathematics, which brought with it the realisation that the paradox of the liar could infect even mathematics itself. This appears most simply in the form of 'Russell's paradox', appropriately in the heart of set theory. At first blush one might think that where sets are concerned any intension has an extension: this is the axiom of naïve set existence. For any property of sets there exists a set containing precisely the sets with that property, all of those and no others. This leads rapidly to Russell's paradox, the paradox of the class of all sets that are not members of themselves. This is the Russell class. Is it a member of itself? Well, if it is, it isn't, and if it isn't, it is. This is Russell's paradox. The aperçu that leapt to mind was that the problem has something to do with the possibility of sets being members of themselves, or to do with defining sets in terms of membership of themselves. Although these two might sound like formulations of the same insight, they nevertheless lead to radically different resolutions, and to two traditions in set theory represented by Zermelo-Fraenkel set theory (often just called 'set theory' by its votaries, and in any case universally abbreviated to 'ZF') and Quine's NF" (ibid., pp. 838-9). 8vo, pp. 538-540. Original self-wrappers. Bestandsnummer des Verkäufers ABE-1609178400555

Bibliografische Details

Titel: On the consistency of 'New Foundations'. ...

Erscheinungsdatum: 1951

Einband: Soft cover

Zustand: Near Fine

Auflage: 1st Edition