Skip to main content
×
×
Home

Pointwise definable models of set theory

  • Joel David Hamkins (a1) (a2), David Linetsky (a3) and Jonas Reitz (a4)
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.

Copyright
References
Hide All
[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.
[BT09]Brooke-Taylor, Andrew, Large cardinals and definable well-orders on the universe, this Journal, vol. 74 (2009), no. 2, pp. 641654.
[Dav82]David, R., Some applications of Jensen's coding theorem. Annals of Mathematical Logic, vol. 22 (1982), no. 2, pp. 177196.
[Ena02]Enayat, Ali, Counting models of set theory, Fundamenta Mathematicae, vol. 174 (2002), no. 1, pp. 2347.
[Ena05]Enayat, Ali, Models of set theory with definable ordinals, Archive of Mathematical Logic, vol. 44 (2005), pp. 363385.
[FHR]Fuchs, Gunter, Hamkins, Joel David, and Reitz, Jonas, Set-theoretic geology, in preparation.
[HJ]Hamkins, Joel David and Johnstone, Thomas, The Resurrection Axioms, in preparation.
[Jec03]Jech, Thomas, Set theory, 3 ed., Springer Monographs in Mathematics, Springer, 2003.
[KS06]Kossak, Roman and Schmerl, James, The structure of models of Peano Arithmetic, Oxford Logic Guides, vol. 50, Oxford University Press, Oxford, 2006.
[Men97]Mendelson, Elliott, An introduction to mathematical logic, vol. 4, Chapman & Hall, London, 1997.
[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.
[Rei06]Reitz, Jonas, The Ground Axiom, Ph.D. thesis, The Graduate Center of the City University of New York, 09 2006.
[Sim74]Simpson, S., Forcing and models of arithmetic, Proceeding of the American Mathematical Society, vol. 43 (1974), pp. 93194.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 18 *
Loading metrics...

Abstract views

Total abstract views: 295 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 19th July 2018. This data will be updated every 24 hours.