# Blog Archives

# Topic Archive: Gödel-Bernays set theory

# Separating Class Determinacy

In his dissertation, Steel showed that both open determinacy and clopen determinacy have a reverse math strength of $\mathsf{ATR}_0$. In particular, this implies that clopen determinacy is equivalent to open determinacy (over a weak base theory). Gitman and Hamkins in recent work explored determinacy for class-sized games in the context of second-order set theory. They asked whether the analogue of Steel’s result holds: over $\mathsf{GBC}$, is open determinacy for class games equivalent to clopen determinacy for class games?

The answer, perhaps surprisingly, is no. In this talk I will present some recent results by Hachtman which answer the question of Gitman and Hamkins. We will see how to construct a transitive model of $\mathsf{GBC}$ which satisfies clopen class determinacy but does not satisfy open class determinacy.

# Minimal models of second-order set theories

Everyone knows that there is a least transitive model of ZFC. Is the same true for second-order set theories? The main result of this talk is that the answer is no for Kelley-Morse set theory. Another notion of minimality we will consider is being the least model with a fixed first-order part. We will see that no countable model of ZFC has a least KM-realization. Along the way, we will look at the analogous questions for Gödel-Bernays set theory.

# Rather classless models of set theory and second-order set theory

A model $M$ of ZFC is rather classless if every class of $M$ all of whose bounded initial segments are in $M$ is definable in $M$. In this talk, we will construct rather classless end extensions for every countable model of set theory. As an application of this construction, we will see that there are models of ZFC with precisely one extension to a model of GBC and that there are models of set theory which admit no extension to a model of GBC. If time permits, we will look at some related constructions with models of KM + the axiom schema of class choice.