@MISC{Forster98quine'snf--60, author = {Thomas Forster}, title = {Quine's NF--60 years on}, year = {1998} }

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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.