Generalizations of the Kunen inconsistency

  • J. D. Hamkins, G. Kirmayer, and N. L. Perlmutter, “Generalizations of the Kunen inconsistency,” Annals of Pure and Applied Logic, vol. 163, iss. 12, pp. 1872-1890, 2012.  
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    title = "Generalizations of the {Kunen} inconsistency",
    journal = "Annals of Pure and Applied Logic",
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    author = "Joel David Hamkins and Greg Kirmayer and Norman Lewis Perlmutter"
    }

We present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself. For example, there is no elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or indeed from any definable class to V, among many other possibilities we consider, including generic embeddings, definable embeddings and results not requiring the axiom of choice. We have aimed in this article for a unified presentation that weaves together some previously known unpublished or folklore results, several due to Woodin and others, along with our new contributions.

Pointwise definable models of set theory

  • J. D. Hamkins, D. Linetsky, and J. Reitz, “Pointwise definable models of set theory,” J. Symbolic Logic, vol. 78, iss. 1, pp. 139-156, 2013.  
    @article {HamkinsLinetskyReitz2013:PointwiseDefinableModelsOfSetTheory,
    AUTHOR = {Hamkins, Joel David and Linetsky, David and Reitz, Jonas},
    TITLE = {Pointwise definable models of set theory},
    JOURNAL = {J. Symbolic Logic},
    FJOURNAL = {Journal of Symbolic Logic},
    VOLUME = {78},
    YEAR = {2013},
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    PAGES = {139--156},
    ISSN = {0022-4812},
    MRCLASS = {03E55},
    MRNUMBER = {3087066},
    MRREVIEWER = {Bernhard A. K{\"o}nig},
    DOI = {10.2178/jsl.7801090},
    URL = {http://jdh.hamkins.org/pointwisedefinablemodelsofsettheory/},
    eprint = "1105.4597",
    archivePrefix = {arXiv},
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One occasionally hears the argument—let us call it the math-tea argument, for perhaps it is heard at a good math tea—that there must be real numbers that we cannot describe or define, because there are are only countably many definitions, but uncountably many reals.  Does it withstand scrutiny?

This article provides an answer.  The article has a dual nature, with the first part aimed at a more general audience, and the second part providing a proof of the main theorem:  every countable model of set theory has an extension in which every set and class is definable without parameters.  The existence of these models therefore exhibit the difficulties in formalizing the math tea argument, and show that robust violations of the math tea argument can occur in virtually any set-theoretic context.

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V=HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Godel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.

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

  • J. D. Hamkins, J. Reitz, and W. Woodin, “The ground axiom is consistent with $V\ne{\rm HOD}$,” Proc.~Amer.~Math.~Soc., vol. 136, iss. 8, pp. 2943-2949, 2008.  
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    AUTHOR = {Hamkins, Joel David and Reitz, Jonas and Woodin, W.~Hugh},
    TITLE = {The ground axiom is consistent with {$V\ne{\rm HOD}$}},
    JOURNAL = {Proc.~Amer.~Math.~Soc.},
    FJOURNAL = {Proceedings of the American Mathematical Society},
    VOLUME = {136},
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    PAGES = {2943--2949},
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    MRREVIEWER = {P{\'e}ter Komj{\'a}th},
    DOI = {10.1090/S0002-9939-08-09285-X},
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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.

Generalizations of the Kunen Inconsistency, Singapore 2011

A talk at the Prospects of Infinity: Workshop on Set Theory  at the National University of Singapore, July 18-22, 2011.

I shall present several generalizations of the well-known Kunen inconsistency that there is no nontrivial elementary embedding from the set-theoretic universe V to itself, including generalizations-of-generalizations previously established by Woodin and others.  For example, there is no nontrivial elementary embedding from the universe V to a set-forcing extension V[G], or conversely from V[G] to V, or more generally from one ground model of the universe to another, or between any two models that are eventually stationary correct, or from V to HOD, or conversely from HOD to V, or from V to the gHOD, or conversely from gHOD to V; indeed, there can be no nontrivial elementary embedding from any definable class to V.  Other results concern generic embeddings, definable embeddings and results not requiring the axiom of choice.  I will aim for a unified presentation that weaves together previously known unpublished or folklore results along with some new contributions.  This is joint work with Greg Kirmayer and Norman Perlmutter.

SlidesArticle 

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

  • J. D. Hamkins, “Pointwise definable models of set theory, extended abstract,” Mathematisches Forschungsinstitut Oberwolfach Report, vol. 8, iss. 1, 02/2011, pp. 128-131, 2011.  
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    AUTHOR = "Joel David Hamkins",
    TITLE = "Pointwise definable models of set theory, extended abstract",
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    volume = "8",
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    DOI = {10.4171/OWR/2011/02},
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This is an extended abstract for the talk I gave at the Mathematisches Forschungsinstitut Oberwolfach, January 9-15, 2011.

Slides | Main Article