Topic Archive: class determinacy

Set theory seminarFriday, November 4, 201610:00 amGC 6417

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.