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Pointwise definable models of set theory

Published online by Cambridge University Press:  12 March 2014

Joel David Hamkins ,
David Linetsky  and
Jonas Reitz
Joel David Hamkins
Affiliation:
The Graduate Center, The city University of New York, 365 Fifth Avenue, New York, NY 10016, USA College of Staten Island, The City University of New York, 2800 Victory Boulevard, Staten Island, NY 10314, USA, E-mail: [email protected] URL: http://jdh.hamkins.org
David Linetsky
Affiliation:
Phreesia Inc., 432 Park Avenue South, 12th Floor, New York, NY 10016, USA, E-mail: [email protected]
Jonas Reitz
Affiliation:
New York City College of Technology, The City University of New York, 300 Jay Street, Brooklyn, NY 11201, USA, E-mail: [email protected]
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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.

Keywords

set theoryforcing
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 1 , March 2013 , pp. 139 - 156
Copyright
Copyright © Association for Symbolic Logic 2013

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