## Quine's NF--60 years on (1998)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Forster98quine'snf--60,

author = {Thomas Forster},

title = {Quine's NF--60 years on},

year = {1998}

}

### OpenURL

### Abstract

Sixty years ago in this journal, the distinguished American philosopher W.V. Quine published a novel approach to set theory. The title was New Foundations for Mathematical Logic [6]. The diamond anniversary is being commemorated by a workshop in Cambridge (England) and comes at a time of rapid increase of interest in the alternatives to the hitherto customary Zermelo-Fr"ankel set theory, which promises a new lease of life for the axiomatic system now known as `NF'; its creator remains in good health too. Although he is best known to a wider public for his philosophical writings, his most enduring and most concrete legacy for the next fifty years may well turn out to be his most mathematical: he gave us NF. Set theory is the study of sets, which are the simplest of all mathematical entities. Let us illustrate by constrasting sets with groups. Two distinct groups can have the same elements and yet be told apart by the way those elements are related. Sets are distinguished from all other mathematical fauna by the fact that a set is constituted solely by its members: two sets with the same members are the same set. To use a bit of jargon from another age, sets are properties in extension. As a result, all set theories have the axiom of extensionality: (8xy)(x = y! (8z)(z 2 x! z 2 y)): they differ in their views on which properties have extensions. Since set theory first sprang on the scene about a hundred years ago there has been a tendency to attempt to use this simplicity to simplify and illuminate the rest of mathematics by translating (perhaps a better word is implementing) it into set theory. After all, if we can represent all of mathematics as facts about these delightfully simple things, some facts about mathematics might become clear that would otherwise remain obscure. This same simplicity means that set theory is always a good topic on which to try out any new mathematical idea.

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(Show Context)
Citation Context ...or this? A fashionable candidate about which a lot has been written recently is ZF with "antifoundation" axioms, of which a racy and entertaining treatment can be found in the recently published book =-=[1]-=-. Antifoundation axioms ensure that all binary relations between mathematical objects of interest are representable by 2 between the sets chosen to implement 7sthose mathematical objects. In a way thi... |

34 |
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(Show Context)
Citation Context ...Forster July 18, 1998 Sixty years ago in this journal, the distinguished American philosopher W.V. Quine published a novel approach to set theory. The title was New Foundations for Mathematical Logic =-=[6]-=-. The diamond anniversary is being commemorated by a workshop in Cambridge (England) and comes at a time of rapid increase of interest in the alternatives to the hitherto customary Zermelo-Fr"ankel se... |

18 |
Elementary Set Theory with a Universal Set, originally volume 10 of the Cahiers du Centre de Logique, Academia, Louvainla-Neuve, Belgium; corrected online version at http://math.boisestate.edu/ ~holmes/holmes/head.ps
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- 1998
(Show Context)
Citation Context ...t theory) and that even though NFU is consistent, NF itself isn't.) But even if we do not yet understand clearly why NFU is so much weaker than NF, we can at least start to put this new system to use =-=[4]-=-. There is for the moment a great interest in alternatives to ZF, driven by the feeling that certain structures with non-wellfounded relations on them ought to be represented by sets. (A relation R on... |

10 |
The axiom of choice in Quine's „New Foundations for Mathematical Logic
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(Show Context)
Citation Context ...ng NF's youth. The first really interesting development did not take place until 1953, when E.P. Specker in Z"urich showed that NF refuted the axiom of choice and thereby proved the axiom of infinity =-=[7]-=-. This result was a most mysterious and disquieting one, best approached in the context of another result of Specker's, nine years later, that is in many ways more illuminating. Specker's 1962 paper [... |

6 |
The set-theoretical program of Quine succeeded, but nobody noticed
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(Show Context)
Citation Context ...rely as a vindication of Quine's insight that the type disciplines are enough by themselves to banish the paradoxes, even if we flirt with danger by playing with a bit of polymorphism, as does Holmes =-=[3]-=-. Although it certainly is such a vindication, it raises bigger questions than it answers. After all, if type disciplines are enough to put paradox to flight even when relaxed with polymorphism, why i... |

6 |
On the consistency of a slight (?) modification of Quine’s nf. Synthese
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(Show Context)
Citation Context ...an those for which it fails. A slightly smoother notion is strongly cantorian. A set x is strongly cantorian if and only if the restriction of the singleton function to x is a set. Theorems of Jensen =-=[5]-=- and Holmes [3] tell us that the hereditarily strongly cantorian sets can be almost any ZF-style model we want. A place to look for substructures of models of NFU in which every set is small and antif... |

1 |
Dialectica 12 (1958) 451\Gamma 65. 9. Specker, E.P. Typical ambiguity
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(Show Context)
Citation Context ...executed as follows: take a formula, increase all the type subscripts in it by 1. The result is a new formula, written `OE+' if the first formula was `OE'. What is the relation between OE and OE+? In =-=[8]-=- Specker drew a parallel with projective geometry, which also has an automorphism like this. By interchanging `point' and `line', and interchanging `lie on' with `meet at' one can transform an asserti... |