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March 2013 Pointwise definable models of set theory
Joel David Hamkins, David Linetsky, Jonas Reitz
J. Symbolic Logic 78(1): 139-156 (March 2013). DOI: 10.2178/jsl.7801090

Abstract

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.

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Joel David Hamkins. David Linetsky. Jonas Reitz. "Pointwise definable models of set theory." J. Symbolic Logic 78 (1) 139 - 156, March 2013. https://doi.org/10.2178/jsl.7801090

Information

Published: March 2013
First available in Project Euclid: 23 January 2013

zbMATH: 1270.03101
MathSciNet: MR3087066
Digital Object Identifier: 10.2178/jsl.7801090

Subjects:
Primary: 03E55

Keywords: Forcing, Set theory

Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 1 • March 2013
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