Pointwise definable models of set theory

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

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Joel David Hamkins
Oxford University

Citations of this work

Algebraicity and Implicit Definability in Set Theory.Joel David Hamkins & Cole Leahy - 2016 - Notre Dame Journal of Formal Logic 57 (3):431-439.
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Typicality À la Russell in Set Theory.Athanassios Tzouvaras - 2022 - Notre Dame Journal of Formal Logic 63 (2).

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References found in this work

[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 - 2005 - Archive for Mathematical Logic 44 (3):363-385.

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