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HOD, V AND THE GCH

  • MOHAMMAD GOLSHANI (a1)

Abstract

Starting from large cardinals we construct a model of ZFC in which the GCH fails everywhere, but such that GCH holds in its HOD. The result answers a question of Sy Friedman. Also, relative to the existence of large cardinals, we produce a model of ZFC + GCH such that GCH fails everywhere in its HOD.

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[1] Brooke-Taylor, A. D., Large Cardinals and definable well-orders on the universe, this Journal, vol. 74 (2009), no. 2, pp. 641654.
[2] Cummings, J., David Friedman, S., and Golshani, M., Collapsing the cardinals of HOD . Journal of Mathematical Logic. vol. 15 (2015), no. 2, 32 pages.
[3] Friedman, S.-D., Personal communication.
[4] Friedman, S.-D. and Golshani, Mohammad, Killing GCH everywhere by a cofinality-preserving forcing notion over a model of GCH . Fundamenta Mathematicae, vol. 223 (2013), no. 2, pp. 171193.
[5] Fuchs, G., David Hamkins, J., and Reitz, J., Set-theoretic geology . Annals of Pure and Applied Logic, vol. 166 (2015), no. 4, pp. 464501.
[6] Gitik, M., Prikry-type forcings , Handbook of Set Theory, Springer, Dordrecht, 2010, pp. 13511447.
[7] Merimovich, C., A power function with a fixed finite gap everywhere, this Journal, vol. 72 (2007), no. 2, pp. 361417.
[8] Roguski, S., The theory of the class HOD , Set Theory and Hierarchy Theory, V (Proc. Third Conf., Bierutowice, 1976), Lecture Notes in Mathematics, vol. 619, Springer, Berlin, 1977, pp. 251255.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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