Separating class determinacy

by kamerynwilliams

This was a talk at the CUNY Set Theory Seminar.

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.