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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Joel David Hamkins, Jonas Reitz and W. Hugh Woodin
Journal: Proc. Amer. Math. Soc. 136 (2008), 2943-2949
MSC (2000): Primary 03E35, 03E45, 03E55
Published electronically: April 15, 2008
MathSciNet review: 2399062
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Abstract | References | Similar Articles | Additional Information

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 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.


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Additional Information

Joel David Hamkins
Affiliation: Mathematics Program, The Graduate Center of The City University of New York, 365 Fifth Avenue, New York, New York 10016–and–Department of Mathematics, The College of Staten Island of The City University of New York, 2800 Victory Boulevard, Staten Island, New York 10314
Email: jhamkins@gc.cuny.edu

Jonas Reitz
Affiliation: Mathematics Program, The Graduate Center of The City University of New York, 365 Fifth Avenue, New York, New York 10016–and–Department of Mathematics, New York City College of Technology of The City University of New York, 300 Jay Street, Brooklyn, New York 11201
Email: jonasreitz@gmail.com

W. Hugh Woodin
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Email: woodin@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09285-X
PII: S 0002-9939(08)09285-X
Received by editor(s): February 22, 2007
Received by editor(s) in revised form: June 11, 2007, and June 26, 2007
Published electronically: April 15, 2008
Additional Notes: The research of the first author has been supported in part by grants from the CUNY Research Foundation and from the Netherlands Organization for Scientific Research.
The research of the third author has been supported in part by the National Science Foundation.
We note that the authors of this article constitute three mathematical generations: Reitz was a dissertation student of Hamkins, who was a dissertation student of Woodin.
Communicated by: Julia Knight
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.