Predicativity and Feferman

Abstract

Predicativity is a notable example of fruitful interaction between philosophy and mathematical logic. It originated at the beginning of the 20th century from methodological and philosophical reflections on a changing concept of set. A clarification of this notion has prompted the development of fundamental new technical instruments, from Russell's type theory to an important chapter in proof theory, which saw the decisive involvement of Kreisel, Feferman and Schütte. The technical outcomes of predicativity have since taken a life of their own, but have also produced a deeper understanding of the notion of predicativity, therefore witnessing the "light logic throws on problems in the foundations of mathematics." Predicativity has been at the center of a considerable part of Feferman's work: over the years he has explored alternative ways of explicating and analyzing this notion and has shown that predicative mathematics extends much further than expected within ordinary mathematics. The aim of this note is to outline the principal features of predicativity, from its original motivations at the start of the past century to its logical analysis in the 1950-1960s. The hope is to convey why predicativity is a fascinating subject, which has attracted Feferman's attention over the years.