Skip to main content
SpringerLink
Log in

Model theory of the regularity and reflection schemes

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF \({(\mathcal{L})}\) (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here \({\mathcal{L}}\) is a language with a distinguished linear order <, and REF \({(\mathcal {L})}\) consists of formulas of the form

$$\exists x \forall y_{1} < x \ldots \forall y_{n} < x \varphi (y_{1},\ldots ,y_{n})\leftrightarrow \varphi^{ < x}(y_1, \ldots ,y_n),$$

where φ is an \({\mathcal{L}}\) -formula, φ <x is the \({\mathcal{L}}\) -formula obtained by restricting all the quantifiers of φ to the initial segment determined by x, and x is a variable that does not appear in φ. Our results include:

Theorem

The following five conditions are equivalent for a complete first order theory T in a countable language \({\mathcal{L}}\) with a distinguished linear order:

  1. (1)

    Some model of T has an elementary end extension with a first new element.

  2. (2)

    T ⊢ REF \({(\mathcal{L})}\).

  3. (3)

    T has an ω 1-like model that continuously embeds ω 1.

  4. (4)

    For some regular uncountable cardinal κ, T has a κ-like model that continuously embeds a stationary subset of κ.

  5. (5)

    For some regular uncountable cardinal κ, T has a κ-like model \({\mathfrak{M}}\) that has an elementary extension in which the supremum of M exists.

Moreover, if κ is a regular cardinal satisfying κ = κ <κ, then each of the above conditions is equivalent to:

  1. (6)

    T has a κ + -like model that continuously embeds a stationary subset of κ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barwise J., Schlipf J.: An introduction to recursively saturated and resplendent models. J. Sym. Logic 41(2), 531–536 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chang C.C.: A note on the two cardinal problem. Proc. Am. Math. Soc. 16, 1148–1155 (1965)

    Article  MATH  Google Scholar 

  3. Chang C.C, Keisler H.J.: Model Theory, Studies in Logic and the Foundations of Mathematics, 3rd edn, vol. 73. North-Holland Publishing Co., Amsterdam (1990)

    Google Scholar 

  4. Easton W.: Powers of regular cardinals. Ann. Math Logic 1, 139–178 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ebbinghaus H.-D.: Extended logics: the general framework. In: Barwise, J., Feferman, S. (eds) Model Theoretic Logics, pp. 25–76. Springer-Verlag, New York (1985)

    Google Scholar 

  6. Ehrenfeucht, A.: An application of games to the completeness problem for formalized theories. Fund. Math. 49, 129–141 (1960/1961)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Enayat A.: Power-like models of set theory. J. Sym. Logic 66, 1766–1782 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  11. Hodges W.: Building Models by Games. London Mathematical Society Student Texts, vol. 2. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  12. Hutchinson J.: Elementary extensions of countable models of set theory. J. Sym. Logic 41(1), 139–145 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jech T.: Set Theory, Springer Monographs in Mathematics. Springer, Berlin (2003)

    Google Scholar 

  14. Jensen R.: The fine structure of the constructible hierarchy. Ann. Math. Logic 4, 229–308 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kanamori A.: Levy and set theory. Ann. Pure Appl. Logic 140, 233–252 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kaufmann M.: Blunt and topless end extensions of models of set theory. J. Sym. Logic 48, 1053–1073 (1984)

    Article  MathSciNet  Google Scholar 

  17. Kaye R.: Models of Peano Arithmetic. Oxford University Press, Oxford (1991)

    MATH  Google Scholar 

  18. Keisler H.J.: Some model theoretic results on ω-logic. Israel J. Math. 4, 249–261 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  19. Keisler H.J.: Model Theory for Infinitary Logic, Studies in Logic and the Foundations of Mathematics, vol. 62. North-Holland Publishing Co., Amsterdam (1971)

    Google Scholar 

  20. Kennedy, J., Väänänen, J.: Applications of regular filters and square principles in model theory, Quaderni di Matematica vol. 17, (A. Andretta, ed.), Department of Mathematics, Seconda Università di Napoli, Caserta, (2005)

  21. Kossak R., Schmerl J.: The Structure of Models of Peano Arithmetic. Oxford University Press, Oxford (2007)

    Google Scholar 

  22. Mills G., Paris J.: Regularity in models of arithmetic. J. Sym. Logic 49, 272–280 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mohsenipour, S.: Elementary End Extensions in Model Theory and Set Theory, Doctoral Dissertation. IPM, Tehran Iran (2005)

  24. Rosenstein J.G.: Linear Orderings. Academic Press, New York (1982)

    MATH  Google Scholar 

  25. Schlipf J.: Toward model theory through recursive saturation. J. Sym. Logic 43, 183–206 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  26. Schmerl J.: Generalizing special Aronszajn trees. J. Sym. Logic 39, 732–740 (1974)

    Article  MathSciNet  Google Scholar 

  27. Schmerl J.: On κ-like structures which embed stationary and closed unbounded subsets. Ann. Math. Logic 10, 289–314 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  28. Schmerl J.: Transfer theorems and their applications to logics. In: Barwise, J., Feferman, S. (eds) Model Theoretic Logics, pp. 177–209. Springer, New York (1985)

    Google Scholar 

  29. Schmerl J., Shelah S.: On power-like models for hyperinaccessible cardinals. J. Sym. Logic 37, 531–537 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  30. Shelah S.: Generalized quantifiers and compact logic. Trans. Am. Math. Soc. 204, 342–364 (1975)

    Article  MATH  Google Scholar 

  31. Shelah S.: End extensions and the number of countable models. J. Sym. Logic 43, 550–562 (1978)

    Article  MATH  Google Scholar 

  32. Shelah S.: “Gap 1” two-cardinal principles and the omitting types theorem for \({\mathcal{L}(Q)}\) . Israel J. Math. 65, 133–152 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  33. Shelah S., Väanänen J.: Recursive logic frames. MLQ Math. Log. Q. 52, 151–164 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Simpson S.G.: Subsystems of Second Order Arithmetic. Springer, Berlin (1999)

    MATH  Google Scholar 

  35. Villegas-Silva L.M.: A gap 1 cardinal transfer theorem. MLQ Math. Log. Q. 2, 340–350 (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, American University, 4400 Mass. Ave. NW., Washington, DC, 20016-8050, USA

    Ali Enayat

  2. School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran

    Shahram Mohsenipour

Authors
  1. Ali Enayat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shahram Mohsenipour
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali Enayat.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Enayat, A., Mohsenipour, S. Model theory of the regularity and reflection schemes. Arch. Math. Logic 47, 447–464 (2008). https://doi.org/10.1007/s00153-008-0089-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-008-0089-z

Mathematics Subject Classification (2000)

Search

Navigation