Why predicative sets?

The predicativist program for the foundations of mathematics, initiated by Poincaré and first developed by Weyl, seeks to establish certainty in mathematics without revolutionizing it. The program was later extensively pursued by Feferman, who developed proofs systems for predicative mathematics, and showed that a very large part of classical analysis can be developed within them. Both Weyl and Feferman worked within type-theoretic frameworks. In contrast, set theory is almost universally accepted now as the foundational theory in which the whole of mathematics can and should be developed. We explain how to reconstruct Weyl’s ideas and system within the set-theoretical framework, and indicate the advantages that this approach to predicativity and to set theory has from both the foundational as well as the computational points of views.