The set-theoretical program of Quine succeeded, but nobody noticed

Abstract

The set theory "New Foundations" or NF introduced by W.V.O. Quine in 1935 is discussed, along with related systems. It is argued that, in spite of the fact that the consistency of NF remains an open question, the relative consistency results for NFU obtained by R. B. Jensen in 1969 demonstrate that Quine's general approach can be used successfully. The development of basic mathematical concepts in a version of NFU is outlined. The interpretation of set theories of the usual type in extensions of NFU is discussed. The problems with NF itself are discussed, and other fragments of NF known to be consistent are briefly introduced. The relative merits of Quine-style and Zermelo-style set theories are considered from a philosophical standpoint. Finally, systems of untyped combinatory logic or $\lambda$-calculus related to NF and its fragments are introduced, and their relation to an abstract model of programming is outlined.