Abstract
We show, assuming weak large cardinals, that in the context of games of length \(\omega \) 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 \((\omega +2)\)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.
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Acknowledgments
I am grateful to Victoria Gitman for introducing me to the questions discussed here. I also thank the American Institute of Mathematics and organizers of the workshop “High and Low Forcing” held in January, 2016, which allowed these initial conversations to take place.
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The author was partially supported in this work by NSF Grant DMS-1246844.
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Hachtman, S. Determinacy separations for class games. Arch. Math. Logic 58, 635–648 (2019). https://doi.org/10.1007/s00153-018-0655-y
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DOI: https://doi.org/10.1007/s00153-018-0655-y