Determinacy separations for class games

TL;DR

This paper demonstrates that, under weak large cardinal assumptions, clopen determinacy is strictly weaker than open determinacy for class games, providing a hierarchy of determinacy principles in various logical settings.

Contribution

It establishes a hierarchy of determinacy principles for class games under large cardinal assumptions, extending the analysis to various logical frameworks and answering open questions.

Findings

Clopen determinacy is weaker than open determinacy under certain assumptions.

Bounds on the strength of Borel determinacy for proper class games.

Applicability of results to various levels of arithmetic and class theories.

Abstract

We show, assuming weak large cardinals, that in the context of games played in a proper class of moves, clopen determinacy is strictly weaker than open determinacy. The proof amounts to an analysis of a certain level of $L$ that exists under large cardinal assumptions weaker than an inaccessible. Our argument is sufficiently general to give a family of determinacy separation results applying in any setting where the universal class is sufficiently closed; e.g., in third, seventh, or $(\omega+2)$th order arithmetic. We also prove bounds on the strength of Borel determinacy for proper class games. These results answer questions of Gitman and Hamkins.