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Determinacy separations for class games

  • Sherwood HachtmanEmail author
Article
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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.

Keywords

Set theory with classes Determinacy Higher order arithmetic Constructibility Admissible set theory 

Mathematics Subject Classification

03E70 03E60 03F35 

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Notes

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

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