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

[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 $V=\text{HOD}$. 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.

Pointwise definable models of set theory, extended abstract, Oberwolfach 2011

[bibtex key=Hamkins2011:PointwiseDefinableModelsOfSetTheoryExtendedAbstract]

This is an extended abstract for the talk I gave at the Mathematisches Forschungsinstitut Oberwolfach, January 9-15, 2011.

Slides | Main Article