This will be a talk on 14th November 2023 at the Konstanz Logic Colloquium. Handout here.

Abstract: On one view of set theory, the paradoxes precipitated a radical clarification of our concept of set, culminating with the isolation of the iterative conception of set. In this talk, I’ll present some work from a book I’ve been working on (also entitled Engineering Set-Theoretic Concepts). I’ll argue that in fact the iterative conception admits of further splitting into multiple conceptions of set, and that we are ourselves at a conceptual crossroads motivated by a kind of paradox. In one direction, we are pushed to the standard picture of ZFC. In the other, we are pushed towards a conception of set on which every set is countable. I’ll also (for the logic folks) present some of the mathematics behind this latter less familiar picture, and situate ZFC-based set theory within it.