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 = HOD. In this article, we show that the Ground Axiom is relatively consistent with V ≠ HOD. In fact, every model of ZFC has a class-forcing extension that is a model of ZFC + GA + V ≠ HOD. The method accommodates large cardinals: every model of ZFC with a supercompact cardinal, for example, has a class-forcing extension with ZFC + GA + V ≠ HOD in which this supercompact cardinal is preserved.
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