Work

The ground axiom is consistent with V $\neq $ HOD

Year: 2008

Type: article

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

Source: Proceedings of the American Mathematical Society

DOI: 10.1090/s0002-9939-08-09285-x

Authors: Joel David Hamkins, Jonas Reitz, W. Hugh Woodin

Institutions: The Graduate Center, CUNY, College of Staten Island, City University of New York, New York City College of Technology, University of California, Berkeley

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funders: National Science Foundation, City University of New York, Research Foundation of The City University of New York

OpenAlex ID: W2029038421