The ground axiom is consistent with $V\ne{\rm HOD}$

  • J. D. Hamkins, J. Reitz, and W. Woodin, “The ground axiom is consistent with $V\ne{\rm HOD}$,” Proc.~Amer.~Math.~Soc., vol. 136, iss. 8, pp. 2943-2949, 2008.  
    AUTHOR = {Hamkins, Joel David and Reitz, Jonas and Woodin, W.~Hugh},
    TITLE = {The ground axiom is consistent with {$V\ne{\rm HOD}$}},
    JOURNAL = {Proc.~Amer.~Math.~Soc.},
    FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {136},
    YEAR = {2008},
    NUMBER = {8},
    PAGES = {2943--2949},
    ISSN = {0002-9939},
    MRCLASS = {03E35 (03E45 03E55)},
    MRNUMBER = {2399062 (2009b:03137)},
    MRREVIEWER = {P{\'e}ter Komj{\'a}th},
    DOI = {10.1090/S0002-9939-08-09285-X},
    URL = {},
    file = F

Abstract. The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of any inner model. Despite the apparent second-order nature of this assertion, it is first-order expressible in set theory. The previously known models of the Ground Axiom all satisfy strong forms of $V=\text{HOD}$. In this article, we show that the Ground Axiom is relatively consistent with $V\neq\text{HOD}$. In fact, every model of ZFC has a class-forcing extension that is a model of $\text{ZFC}+\text{GA}+V\neq\text{HOD}$. The method accommodates large cardinals: every model of ZFC with a supercompact cardinal, for example, has a class-forcing extension with $\text{ZFC}+\text{GA}+V\neq\text{HOD}$ in which this supercompact cardinal is preserved.

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