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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The ground axiom is consistent with V $\neq$ HOD
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by Joel David Hamkins, Jonas Reitz and W. Hugh Woodin PDF
Proc. Amer. Math. Soc. 136 (2008), 2943-2949 Request permission

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.
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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
  • MR Author ID: 347679
  • 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
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 2943-2949
  • MSC (2000): Primary 03E35, 03E45, 03E55
  • DOI: https://doi.org/10.1090/S0002-9939-08-09285-X
  • MathSciNet review: 2399062