Set theory in second-order

This is a talk at the Set Theory in UK 2 meeting, University of Bristol, May 8, 2019.

Abstract: Classes, from class forcing notions to elementary embeddings of the universe to inner models, play a fundamental role in modern set theory. But within first-order set theory we are limited to studying only definable classes and we cannot even express properties that necessitate quantifying over classes. Second-order set theory is a formal framework in which a model consists both of a collection of sets and a collection of classes (which are themselves collections of sets). In second-order set theory, we can study classes such as truth predicates, which can never be definable over a model of ${\rm ZFC}$, and properties that, for instance, quantify over all inner models. With this formal background we can develop a theory of class forcing that explains why and when class forcing behaves differently from set forcing. In this talk, I will discuss a hierarchy of second-order set theories, starting from the weak Gödel-Bernays set theory ${\rm GBC}$ and going beyond the relatively strong Kelley-Morse theory ${\rm KM}$. I will give an overview of a number of interesting second-order set theoretic principles that arose out of recent work in this area, such as, class choice principles, transfinite recursion with classes, determinacy of games on the ordinals, and the class Fodor Principle. The study of where these principles fit in the hierarchy of second-order set theories should serve as the beginning of a reverse mathematics program that I hope this talk will encourage set theorists to take part in.