@MISC{Parsons_strictpredicativity, author = {Charles Parsons}, title = {STRICT PREDICATIVITY 1}, year = {} }

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Abstract

The most basic notion of impredicativity applies to specifications or definitions of sets or classes. If a set b is specified as {x: A(x)} for some predicate A, then the specification is impredicative if A contains quantifiers such that the set b falls in the range of these quantifiers. Already in the early history of the notion, it was extended to other cases, such as propositions in Russell’s discussions of the liar paradox. Mathematics will be predicative if it avoids impredicative definitions. The logicist reduction of the concept of natural number met a difficulty on this point, since the definition of ‘natural number ’ already given in the work of Frege and Dedekind is impredicative. More recently, it has been argued by Michael Dummett, the author, and Edward Nelson that more informal explanations of the concept of natural number are impredicative as well. That has the consequence that impredicativity is more pervasive in mathematics, and appears at lower levels, than the earlier debates about the issue generally presupposed. The appearance to the contrary resulted historically from the fact that many opponents of impredicative methods, in particular Poincaré and Weyl, were prepared to assume the natural numbers in their work. In this they were followed by later analysts of predicativity, in particular Kreisel, Schütte, and Feferman. The result was that the working conception of predicativity was of predicativity given the natural numbers. Thus Feferman characterized “the predicative conception ” as holding that “only the natural numbers can be