Advertisement

Israel Journal of Mathematics

, Volume 194, Issue 1, pp 409–425 | Cite as

Externally definable sets and dependent pairs

  • Artem Chernikov
  • Pierre Simon
Article

Abstract

We prove that externally definable sets in first order NIP theories have honest definitions, giving a new proof of Shelah’s expansion theorem. Also we discuss a weak notion of stable embeddedness true in this context. Those results are then used to prove a general theorem on dependent pairs, which in particular answers a question of Baldwin and Benedikt on naming an indiscernible sequence.

Keywords

Dependent Theory Dense Pair Unary Predicate Dependent Pair Elementary Extension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Adl08]
    H. Adler, An introduction to theories without the independence property, Archive for Mathematical Logic, to appear.Google Scholar
  2. [BP98]
    Y. Baisalov and B. Poizat, Paires de structures o-minimales, Journal of Symbolic Logic 63 (1998), 570–578.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [BB04]
    B. Baizhanov and J. Baldwin, Local homogeneity, Journal of Symbolic Logic 69 (2004), 1243–1260.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [BB00]
    J. Baldwin and M. Benedikt, Stability theory, permutations of indiscernibles, and embedded finite models, Transactions of the American Mathematical Society 352 (2000), 4937–4969.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [Ber]
    A. Berenstein, Lovely pairs and dense pairs of o-minimal structures, submitted.Google Scholar
  6. [BDO08]
    A. Berenstein, A. Dolich and A. Onshuus, The independence property in generalized dense pairs of structures, Journal of Symbolic Logic 76 (2011), 391–404.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Box09]
    G. Boxall, NIP for some pair-like theories, Archive for Mathematical Logic 50 (2011), 353–359.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [CZ01]
    E. Casanovas and M. Ziegler, Stable theories with a new preicate, Journal of Symbolic logic 66 (2001), 1127–1140.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [vdD98]
    L. van den Dries, Dense pairs of o-minimal structures, Fundamenta Mathematicae 157 (1998), 61–78.MathSciNetzbMATHGoogle Scholar
  10. [vdDL95]
    L. van den Dries and A. H. Lewenberg, T-convexity and tame extensions, Journal of Symbolic Logic 155 (1995), 807–836.Google Scholar
  11. [EP05]
    A. J. Engler and A. Prestel, Valued Dields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005.Google Scholar
  12. [Goo09]
    J. Goodrick, A monotonicity theorem for dp-minimal densely ordered groups, Journal of Symbolic Logic, 2009, to appear.Google Scholar
  13. [Gui09]
    V. Guingona, Dependence and isolated extensions, Proceedings of the American Mathematical Society 139 (2011), 3349–3357.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [GH09]
    A. Günaydin and P. Hieronymi, The real field with the rational points of an elliptic curve, Fundamenta Mathematicae 211 (2011), 15–40.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [GH10]
    A. Günaydin and P. Hieronymi, Dependent pairs, Journal of Symbolic Logic 76 (2011), 377–390.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [Hod93]
    W. Hodges, Model Theory, Encyclopedia of Mathematics and its Applications, Vol. 42, Cambridge University Press, Great Britain, 1993.Google Scholar
  17. [OP07]
    A. Onshuus and Y. Peterzil, A note on stable sets, groups, and theories with nip, Mathematical Logic Quarterly 53 (2007), 295–300.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [Pil07]
    A. Pillay, On externally definable sets and a theorem of Shelah, Felgner Festchrift, Studies in Logic, College Publications, 2007.Google Scholar
  19. [Poi83]
    B. Poizat, Paires de structures stables, The Journal of Symbolic Logic 48 (1983), 239–249.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [She04]
    S. Shelah, Dependent first order theories, continued, Israel Journal of Mathematics 173 (2009), 1–60.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [She05]
    S. Shelah, Strongly dependent theories, 2005, arXiv:math.LO/0504197.Google Scholar

Copyright information

© Hebrew University Magnes Press 2012

Authors and Affiliations

  1. 1.Institut Camille JordanUniversité Claude Bernard — Lyon 1Villeurbanne CedexFrance
  2. 2.Département de mathématiquesEcole Normale SupérieureParis Cedex 05France

Personalised recommendations