Determinacy separations for class games
- 2 Downloads
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.
KeywordsSet theory with classes Determinacy Higher order arithmetic Constructibility Admissible set theory
Mathematics Subject Classification03E70 03E60 03F35
Unable to display preview. Download preview PDF.
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.
- 2.Friedman, H.M.: Higher set theory and mathematical practice. Ann. Math. Log. 2(3), 325–357 (1970/1971)Google Scholar
- 6.Jech, T.: Set Theory. The Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer, Berlin (2003)Google Scholar
- 13.Steel, J.R.: Determinateness and subsystems of analysis. ProQuest LLC, Ann Arbor, MI (1977). Thesis (Ph.D.), University of California, BerkeleyGoogle Scholar