This was a talk at Speaking the Unspeakable: Paradoxes between Truth and Proof at the University of Campinas, Brazil. You can find the slides here.

Abstract:

It is standard in set theory to assume that Cantor’s Theorem establishes that there are uncountable sets. In this paper, we present versions of set theory with classes that imply that every set is countable, and the continuum is a proper class. Within these theories we show how standard set theories (including ZFC with large cardinals added) can be incorporated. We discuss some properties of the theories, in particular that they provide a radically new perspective on the notion of maximality. We conclude that the systems considered raise questions concerning the foundational purpose of set theory.