Journal of Symbolic Logic 78 (1):139-156 (2013)

Authors
Joel David Hamkins
Oxford University
Abstract
A pointwise definable model is one in which every object is \loos 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 theory   forcing
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DOI 10.2178/jsl.7801090
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References found in this work BETA

[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
Set Theory.Keith J. Devlin - 1981 - Journal of Symbolic Logic 46 (4):876-877.
The Ground Axiom.Jonas Reitz - 2007 - Journal of Symbolic Logic 72 (4):1299 - 1317.
Models of Set Theory with Definable Ordinals.Ali Enayat - 2004 - Archive for Mathematical Logic 44 (3):363-385.

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Citations of this work BETA

Ehrenfeucht’s Lemma in Set Theory.Gunter Fuchs, Victoria Gitman & Joel David Hamkins - 2018 - Notre Dame Journal of Formal Logic 59 (3):355-370.

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