The ground axiom is consistent with VHOD

[bibtex key=HamkinsReitzWoodin2008:TheGroundAxiomAndVequalsHOD]

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 VHOD. In fact, every model of ZFC has a class-forcing extension that is a model of ZFC+GA+VHOD. The method accommodates large cardinals: every model of ZFC with a supercompact cardinal, for example, has a class-forcing extension with ZFC+GA+VHOD in which this supercompact cardinal is preserved.

Leave a Reply

Your email address will not be published. Required fields are marked *