Poincaré–weyl’s Predicativity: Going Beyond

Abstract

On the basis of Poincaré and Weyl's view of predicativity as invariance, we develop an extensive framework for predicative, type-free first-order set theory in which Γ 0 and much bigger ordinals can be defined as von Neumann ordinals. This refutes the accepted view of Γ 0 as the "limit of predicativity". §1. Introduction.1.1. What predicativism, and why? In [44] the basic historic problem of the research in foundations of mathematics (FOM) is formulated as follows:How to reconstruct mathematics on a secure basis, one maximally immune to rational doubts.The predicativist program [11, 12, 24,49, 51] has been one of the attempts to solve this basic problem of FOM. It seeks to establish certainty in mathematics without revolutionizing it or changing its underlying classical logic (as the intuitionistic program does). The program was initiated by Poincaré [36][37][38][39]. Its viability was demonstrated by Weyl, who seriously developed it for the first time in his famous small book "Das Kontinuum" [52, 54]. Weyl, and then Feferman [22, 25], have shown that a very large part of classical analysis can be developed within their predicative systems. Feferman further argued that predicative mathematics in fact suffices for developing all the mathematics that is actually indispensable to present-day natural sciences. Hence the predicativist program has been successful in solving the basic problem of FOM. (In my opinion it is the only one about which this can truly be said.)Poincaré's predicativism started as a reaction to the set-theoretical paradoxes. However, in the writings of both Poincaré and Weyl, predicativity derives not so much from the need to avoid paradoxes, but from their