Following his discovery of the paradoxes present in naive set theory, Russell proposed to ban the vicious circle principle, nowadays called impredicative definition. A set is impredicative if its definition quantifies over the totality of sets it belongs to, like the definition of supremum as the least upper bound. Russell's proposal was taken up by Weyl and Feferman in their development of the foundational program predicativist mathematics. The system \(\Pi_1^1-CA_0\) (resp. arithmetical comprehension \(ACA_{0}\)) is a textbook example of impredicative (resp. predicative) mathematics. In this paper, we show that \(\Pi_1^1-CA_0\) can be viewed as an instance of nonstandard arithmetical comprehension. We also show that bar recursion, another inherently impredicative operation, may be viewed as the highly predicative notion of primitive recursion, but with nonstandard numbers. In other words, predicativism seems to be contingent on whether the framework at hand accommodates Nonstandard Analysis, arguably an undesirable feature for a foundational philosophy.

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School of Mathematics

University of Birmingham

http://web.mat.bham.ac.uk/R.W.Kaye/

Dr Walter Dean

Department of Philosophy

University of Warwick

http://go.warwick.ac.uk/whdean