# 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.
@ARTICLE{HamkinsReitzWoodin2008:TheGroundAxiomAndVequalsHOD,
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},
CODEN = {PAMYAR},
MRCLASS = {03E35 (03E45 03E55)},
MRNUMBER = {2399062 (2009b:03137)},
MRREVIEWER = {P{\'e}ter Komj{\'a}th},
DOI = {10.1090/S0002-9939-08-09285-X},
URL = {http://wp.me/p5M0LV-3j},
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.