| Summary: | We show, assuming weak large cardinals, that in the context of games of length ω with moves coming from a proper class, 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 (ω+ 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. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
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