- 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, p. 2943–2949, 2008.

[Bibtex]`@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.