Pointwise definable models of set theory

  • J. D. Hamkins, D. Linetsky, and J. Reitz, “Pointwise definable models of set theory,” Journal of symbolic logic, vol. 78, iss. 1, pp. 139-156, 2013.  
    AUTHOR = "Joel David Hamkins and David Linetsky and Jonas Reitz",
    TITLE = "Pointwise definable models of set theory",
    JOURNAL = "Journal of Symbolic Logic",
    YEAR = "2013",
    volume = "78",
    number = "1",
    pages = "139--156",
    month = "",
    note = "",
    url = "http://projecteuclid.org/euclid.jsl/1358951104",
    abstract = "",
    keywords = "",
    source = "",
    eprint = "1105.4597",
    doi = "10.2178/jsl.7801090",
    file = F

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.

4 thoughts on “Pointwise definable models of set theory

  1. Pingback: Pointwise definable models of set theory, extended abstract, Oberwolfach 2011

  2. Pingback: Must there be non-definable numbers? Pointwise definability and the math tea argument, KGRC, Vienna 2011 | Joel David Hamkins

  3. Pingback: Must there be numbers we cannot describe or define? Pointwise definability and the Math Tea argument, Bristol, April 2012 | Joel David Hamkins

  4. Regarding Theorem 11, i.e. “Every Countable model of ZFC has a pointwise definable class forcing extension”, can one, using Boolean-valued models show that every model of ZFC has a pointwise definable class forcing extension? Also, in the proof of Theorem 11 ( since HOD is being forced) does the Leibniz-Myceielski axiom also hold in the class forcing extension and can the model constructed in Theorem 11 satisfy not-CH (since in the proof the construction forces failures of the GCH)?

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