Determinacy separations for class games

  • Sherwood HachtmanEmail author


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.


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

Mathematics Subject Classification

03E70 03E60 03F35 


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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.


  1. 1.
    Barwise, J.: Admissible Sets and Structures: An Approach to Definability Theory, Perspectives in Mathematical Logic. Springer, Berlin (1975)CrossRefGoogle Scholar
  2. 2.
    Friedman, H.M.: Higher set theory and mathematical practice. Ann. Math. Log. 2(3), 325–357 (1970/1971)Google Scholar
  3. 3.
    Gitman, V., Hamkins, J.D.: Open determinacy for class games. In: Caicedo, A.E., Cummings, J., Koellner, P., Larson, P. (eds.) Foundations of Mathematics. Contemporary Mathematics, vol. 690, pp. 121–143. American Mathematical Society, Providence, RI (2017)CrossRefGoogle Scholar
  4. 4.
    Hachtman, S.: Calibrating determinacy strength in levels of the Borel hierarchy. J. Symb. Log. 82(2), 510–548 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hachtman, S.: Determinacy in third order arithmetic. Ann. Pure Appl. Log. 168(11), 2008–2021 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jech, T.: Set Theory. The Third Millennium Edition, Revised and Expanded. Springer Monographs in Mathematics. Springer, Berlin (2003)Google Scholar
  7. 7.
    Jensen, R.B.: The fine structure of the constructible hierarchy. Ann. Math. Log. 4, 229–308 (1972). [erratum, ibid. 4 (1972), 443 (1972). With a section by Jack Silver]MathSciNetCrossRefGoogle Scholar
  8. 8.
    Mendelson, E.: Introduction to Mathematical Logic. Textbooks in Mathematics, 6th edn. CRC Press, Boca Raton, FL (2015)zbMATHGoogle Scholar
  9. 9.
    Montalbán, A., Shore, R.A.: The limits of determinacy in second-order arithmetic. Proc. Lond. Math. Soc. (3) 104(2), 223–252 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Montalbán, A., Shore, R.A.: The limits of determinacy in second order arithmetic: consistency and complexity strength. Israel J. Math. 204(1), 477–508 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Schindler, R., Zeman, M.: Fine structure. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vol. 1, 2, 3, pp. 605–656. Springer, Dordrecht (2010)CrossRefGoogle Scholar
  12. 12.
    Schweber, N.: Transfinite recursion in higher reverse mathematics. J. Symb. Log. 80(3), 940–969 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Steel, J.R.: Determinateness and subsystems of analysis. ProQuest LLC, Ann Arbor, MI (1977). Thesis (Ph.D.), University of California, BerkeleyGoogle Scholar

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© 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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