Skip to main content

Models of set theory with definable ordinals


A DO model (here also referred to a Paris model) is a model of set theory all of whose ordinals are first order definable in . Jeffrey Paris (1973) initiated the study of DO models and showed that (1) every consistent extension T of ZF has a DO model, and (2) for complete extensions T, T has a unique DO model up to isomorphism iff T proves V=OD. Here we provide a comprehensive treatment of Paris models. Our results include the following:

1. If T is a consistent completion of ZF+VOD, then T has continuum-many countable nonisomorphic Paris models.

2. Every countable model of ZFC has a Paris generic extension.

3. If there is an uncountable well-founded model of ZFC, then for every infinite cardinal κ there is a Paris model of ZF of cardinality κ which has a nontrivial automorphism.

4. For a model ZF, is a prime model ⇒ is a Paris model and satisfies AC is a minimal model. Moreover, Neither implication reverses assuming Con(ZF).

This is a preview of subscription content, access via your institution.


  1. Barwise, J.: Admissible Sets and Structures. Springer-Verlag, Berlin, 1975

  2. Chang, C.C., Keisler, H.J.: Model Theory. Elsevier North Holland, Amsterdam, 1973

  3. Cohen, P.: Set Theory and the Continuum Hypothesis. Benjamin, New York, 1966

  4. Cohen, P.: Automorphisms of set theory. In: Proceedings of the Tarski Symposium, (Proc. Sympos. Pure Math. Vol XXV, Univ. California, Berkeley, Calif. 1971), Amer. Math. Soc. Providence, RI, 1974, pp. 325–330

  5. Enayat, A.: On certain elementary extensions of models of set theory. Trans. Am. Math. Soc. 283, 705–715 (1984)

    MathSciNet  MATH  Google Scholar 

  6. Enayat, A.: Undefinable classes and definable elements in models of set theory and arithmetic. Proc. Am. Math. Soc. 103, 1216–1220 (1988)

    MathSciNet  MATH  Google Scholar 

  7. Enayat, A.: Counting models of set theory. Fund. Math. 174(1), 23–47 (2002)

    MATH  Google Scholar 

  8. Enayat, A.: On the Leibniz-Mycielski axiom in set theory. Fund. Math. 181, 215–231 (2004)

    Google Scholar 

  9. Enayat, A.: Leibnizian Models of Set Theory. J. Sym. Logic 69, 775–789 (2004)

    Google Scholar 

  10. Felgner, U.: Comparisons of the axioms of local and universal choice. Fund. Math. 71, 43–62 (1971)

    MATH  Google Scholar 

  11. Friedman, H.: Large models of countable height. Tran. Am. Math. Soc. 201, 227–239 (1975)

    Google Scholar 

  12. Grigorieff, S.: Intermediate submodels and generic extensions in set theory. Ann. Math. 101, 447–490 (1975)

    MATH  Google Scholar 

  13. Halpern, J.D.: Lévy, A.: The Boolean prime ideal theorem does not imply the axiom of choice. In: Axiomatic Set Theory, D. Scott (ed.), Part I, Proc. Symp. Pure Math. 13, vol. I, American Mathematical Society. Providence, RI, 1971, pp. 83–134

  14. Jech, T.: Set Theory. Academic Press, New York, 1978

  15. Jensen, R., Solovay, R.: Applications of almost disjoint forcing. In: Mathematical Logic and Foundations of Set Theory, Y. Bar-Hillel (ed.), North-Holland, Amsterdam, 1970, pp. 84–104

  16. Keisler, H.J.: Model Theory for Infinitary Logic. North-Holland, Amsterdam, 1971

  17. Kreisel, G., Wang, H.: Some applications of formal consistency proofs. Fund. Math. 42, 101–110 (1955)

    MathSciNet  MATH  Google Scholar 

  18. Kunen, K.: Set Theory. North Holland, Amsterdam, 1980

  19. McAloon, K.: Consistency results about ordinal definability. Annals Math. Logic, 2, 449–467 (1971)

    Google Scholar 

  20. Morley, M.: The number of countable models. J. Sym. Logic, 35, 14–18 (1970)

    Google Scholar 

  21. Mycielski, J.: New set-theoretic axioms derived from a lean metamathematics. J. Sym. Logic, 60, 191–198 (1995)

    Google Scholar 

  22. Mycielski, J.: Axioms which imply GCH. Fund. Math. 176, 193–207 (2003)

    MathSciNet  MATH  Google Scholar 

  23. Mycielski, J., Swierczkowski, S.: On the Lebesgue measurability and the axiom of determinateness. Fund. Math. 54, 67–71 (1964)

    MATH  Google Scholar 

  24. Myhill, J., Scott, D.: Ordinal definability. In: Axiomatic Set Theory, D. Scott (ed.), Part I, Proc. Symp. Pure Math. 13, vol. I, American Mathematical. Society, Providence, RI, 1970, pp. 271–278

  25. Paris, J.: Minimal models of ZF. Proceedings of the Bertrand Russell Memorial Conference, Leeds Univ. Press, 1973, pp. 327–331

  26. Shelah, S.: Can you take Solovay’s inaccessible away? Israel J. Math. 48, 1–47 (1984)

    MathSciNet  MATH  Google Scholar 

  27. Simpson, S.: Forcing and models of arithmetic. Proc. Am. Math. Soc. 43, 93–194 (1974)

    Google Scholar 

  28. Solovay, R.: A model of set theory in which every set of reals is Lebesgue measurable. Ann. Math. 92, 1–56 (1970)

    MATH  Google Scholar 

  29. Suzuki, Y., Wilmers, G.: Non-standard models for set theory. Proceedings of the Bertrand Russell Memorial Conference, Leeds Univ. Press, 1973, pp. 278–314

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ali Enayat.

Additional information

Mathematics Subject Classification (2000): 03C62, 03C50, Secondary 03H99

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Enayat, A. Models of set theory with definable ordinals. Arch. Math. Logic 44, 363–385 (2005).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:


  • Paris Model
  • Mathematical Logic
  • Minimal Model
  • Prime Model
  • Comprehensive Treatment