Graduate Student Seminar- Predicativity: Frege, Philosophy, and Flaws in the Foundations of Mathematics
Wed, 19 April, 2023
4:00pm - 5:00pm
Date and Time: Wednesday, April 19th 4-5 p.m.
Place: Rome 206
Speaker: Keshav Srinivasan, GWU
Title: Predicativity: Frege, Philosophy, and Flaws in the Foundations of Mathematics
Abstract: We will discuss predicativism, a philosophy of mathematics fastidious about avoiding circular definitions. Mathematics today is founded on the Zermelo-Fraenkel axioms of set theory (ZFC), but some philosophers argue that these axioms impermissibly allow for definitions of mathematical objects which quantify over a totality containing the object being defined. We will explain how mathematics can be put on a predicative foundation via Bertrand Russell’s ramified theory of types, but also how that constrains what statements of mathematics are provable. In the final part of the talk, we will review work by Nelson, Burgess, and Hazen about predicative proofs of theorems of Peano arithmetic most mathematicians take for granted, like “the sum of two natural numbers is a natural number” and “a natural number raised to the power of a natural number is a natural number.”