Speaker: Edward N. Zalta
Abstract: In our paper "Number Theory and Infinity Without Mathematics” (Journal of Philosophical Logic, 53 (2024): 1161-1197, doi:10.1007/s10992-024-09762-7), Uri Nodelman and I answer the questions: (1) Which set or number existence axioms are needed to prove the theorems of "ordinary" mathematics? (2) How should Frege's theory of numbers be adapted to a modal setting, so that the fact that equivalence classes of equinumerous properties vary from world to world won't give rise to different numbers at different worlds? (3) Can one reconstruct Frege's theory of numbers in a non-modal setting *without* mathematical primitives such as "the number of Fs” (#F) or mathematical axioms such as Hume's Principle? In this talk, I’ll review the technical developments and our answers (surprisingly, our answer to (1) is “none”) and then draw a number of observations how our results preserve Frege’s attempt to provide logical and epistemological foundations for arithmetic.
Authors: Edward N. Zalta and Uri Nodelman