Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-02T20:02:39.888Z Has data issue: false hasContentIssue false

Pointwise definable models of set theory

Published online by Cambridge University Press:  12 March 2014

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]

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Ani]Anixx, Username, Is analysis as taught in universities in fact the analysis of definable numbers?, MathOverflow, http://mathoverflow.net/questions/44102, 2010-10-29.Google Scholar
[BT09]Brooke-Taylor, Andrew, Large cardinals and definable well-orders on the universe, this Journal, vol. 74 (2009), no. 2, pp. 641654.Google Scholar
[Dav82]David, R., Some applications of Jensen's coding theorem. Annals of Mathematical Logic, vol. 22 (1982), no. 2, pp. 177196.CrossRefGoogle Scholar
[Ena02]Enayat, Ali, Counting models of set theory, Fundamenta Mathematicae, vol. 174 (2002), no. 1, pp. 2347.CrossRefGoogle Scholar
[Ena05]Enayat, Ali, Models of set theory with definable ordinals, Archive of Mathematical Logic, vol. 44 (2005), pp. 363385.CrossRefGoogle Scholar
[FHR]Fuchs, Gunter, Hamkins, Joel David, and Reitz, Jonas, Set-theoretic geology, in preparation.Google Scholar
[HJ]Hamkins, Joel David and Johnstone, Thomas, The Resurrection Axioms, in preparation.Google Scholar
[Jec03]Jech, Thomas, Set theory, 3 ed., Springer Monographs in Mathematics, Springer, 2003.Google Scholar
[KS06]Kossak, Roman and Schmerl, James, The structure of models of Peano Arithmetic, Oxford Logic Guides, vol. 50, Oxford University Press, Oxford, 2006.CrossRefGoogle Scholar
[Men97]Mendelson, Elliott, An introduction to mathematical logic, vol. 4, Chapman & Hall, London, 1997.Google Scholar
[Myh52]Myhill, John, The hypothesis that all classes are nameable, Proceedings of the National Academy of Sciences of the United States of America, vol. 38 (1952), pp. 979981.CrossRefGoogle ScholarPubMed
[Rei06]Reitz, Jonas, The Ground Axiom, Ph.D. thesis, The Graduate Center of the City University of New York, 09 2006.Google Scholar
[Sim74]Simpson, S., Forcing and models of arithmetic, Proceeding of the American Mathematical Society, vol. 43 (1974), pp. 93194.Google Scholar