Must there be numbers we cannot describe or define? Pointwise definability and the Math Tea argument, Bristol, April 2012

This is a talk I plan to give to the set theory seminar at the University of Bristol on April 18, 2012.

An old argument, heard at a good math tea, proceeds: “there must be some real numbers that we can neither describe nor define, since there are uncountably many reals, but only countably many definitions.” Does it withstand scrutiny? In this talk, I will discuss the phenomenon of pointwise definable models of set theory, in which every object is definable without parameters. In addition to classical and folklore results on the existence of pointwise definable models of set theory, the main new theorem is that every countable model of ZFC and indeed of GBC has an extension to a model of set theory with the same ordinals, in which every set and class is definable without parameters. This is joint work with Jonas Reitz and David Linetsky, and builds on work of S. Simpson, R. Kossak, J. Schmerl, S. Friedman and A. Enayat.

slides | article

Must there be non-definable numbers? Pointwise definability and the math-tea argument, KGRC, Vienna 2011

This talk will be a part of the “Advanced Introduction” series for graduate students at the the Kurt Gödel Research Center, November 4, 2011.

An old argument, heard perhaps at math tea, proceeds: “there must be some real numbers that we can neither describe nor define, since there are uncountably many reals, but only countably many definitions.” Does it withstand scrutiny? In this talk, I will discuss the phenomenon of pointwise definable models of set theory, in which every object is definable without parameters. In addition to classical and folklore results on the existence of pointwise definable models of set theory, the main new theorem is that every countable model of ZFC and indeed of GBC has an extension to a model of set theory with the same ordinals, in which every set and class is definable without parameters. This is joint work with Jonas Reitz and David Linetsky, and builds on work of S. Simpson, R. Kossak, J. Schmerl, S. Friedman and A. Enayat.

Slides | Article

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},
    NUMBER = {1},
    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},
    primaryClass = {math.LO},
    }

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.  
    @ARTICLE{HamkinsReitzWoodin2008:TheGroundAxiomAndVequalsHOD,
    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},
    YEAR = {2008},
    NUMBER = {8},
    PAGES = {2943--2949},
    ISSN = {0002-9939},
    CODEN = {PAMYAR},
    MRCLASS = {03E35 (03E45 03E55)},
    MRNUMBER = {2399062 (2009b:03137)},
    MRREVIEWER = {P{\'e}ter Komj{\'a}th},
    DOI = {10.1090/S0002-9939-08-09285-X},
    URL = {http://dx.doi.org/10.1090/S0002-9939-08-09285-X},
    file = F
    }

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

  • 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.  
    @ARTICLE{Hamkins2011:PointwiseDefinableModelsOfSetTheoryExtendedAbstract,
    AUTHOR = "Joel David Hamkins",
    TITLE = "Pointwise definable models of set theory, extended abstract",
    JOURNAL = "Mathematisches Forschungsinstitut Oberwolfach Report",
    YEAR = "2011",
    volume = "8",
    number = "1, 02/2011",
    pages = "128--131",
    month = "",
    note = "",
    abstract = "",
    keywords = "",
    source = "",
    DOI = {10.4171/OWR/2011/02},
    URL = {http://dx.doi.org/10.4171/OWR/2011/02},
    file = F
    }

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

Slides | Main Article