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

Joel David Hamkins
Oxford University
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
Categories (categorize this paper)
DOI 10.2178/jsl.7801090
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive

Upload a copy of this paper     Check publisher's policy     Papers currently archived: 50,147
Through your library

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.

View all 6 references / Add more references

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.

Add more citations

Similar books and articles

Pointwise Definable Substructures of Models of Peano Arithmetic.Roman Murawski - 1988 - Notre Dame Journal of Formal Logic 29 (3):295-308.
Ranking Functions and Rankings on Languages.Franz Huber - 2006 - Artificial Intelligence 170:462-471.
On the Type-Definability of the Binding Group in Simple Theories.Bradd Hart & Ziv Shami - 2005 - Journal of Symbolic Logic 70 (2):379 - 388.
Leibnizian Models of Set Theory.Ali Enayat - 2004 - Journal of Symbolic Logic 69 (3):775-789.
Isomorphisms and Nonisomorphisms of Graph Models.Harold Schellinx - 1991 - Journal of Symbolic Logic 56 (1):227-249.
A Brief Introduction to Algebraic Set Theory.Steve Awodey - 2008 - Bulletin of Symbolic Logic 14 (3):281-298.
The Complexity of the Core Model.William J. Mitchell - 1998 - Journal of Symbolic Logic 63 (4):1393-1398.


Added to PP index

Total views
41 ( #228,453 of 2,324,615 )

Recent downloads (6 months)
2 ( #461,830 of 2,324,615 )

How can I increase my downloads?


My notes