Set theories with classes

This is a talk at the special set theory session, Association for Symbolic Logic 2019 Annual North American Meeting, CUNY Graduate Center, May 20-23, 2019.
Slides

Abstract: Second-order set theory is the set-theoretic cousin of the extensively studied field of second-order arithmetic. It is formalized in a two-sorted logic with separate objects for sets and classes. Second-order set theory removes the limitation imposed by ${\rm ZFC}$ of having to work only with definable collections, and allows us to formally consider interesting never definable classes such as truth predicates. With this formal background, we can study properties of class forcing and understand how and why it fails to share many of the nice properties of set forcing. In this talk, I will discuss the hierarchy of second-order set theories from Gödel-Bernays set theory ${\rm GBC}$ to beyond Kelley-Morse theory ${\rm KM}$. Along the way, I will establish some unexpected connections between models of second-order set theories and models of ${\rm ZFC}$ without the powerset axiom. I will also introduce an incipient program of reverse mathematics of second-order set theory which aims to place in the hierarchy various natural class set-theoretic principles such as the class forcing theorem, transfinite recursion with classes, and determinacy of games on the ordinals.