Journal of Symbolic Logic

Pointwise definable models of set theory

Joel David Hamkins, David Linetsky, and Jonas Reitz

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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 Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.

Article information

J. Symbolic Logic, Volume 78, Issue 1 (2013), 139-156.

First available in Project Euclid: 23 January 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E55: Large cardinals

set theory forcing


Hamkins, Joel David; Linetsky, David; Reitz, Jonas. Pointwise definable models of set theory. J. Symbolic Logic 78 (2013), no. 1, 139--156. doi:10.2178/jsl.7801090.

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