Israel Journal of Mathematics

, Volume 220, Issue 2, pp 947–1014 | Cite as

Model-theoretic applications of cofinality spectrum problems

Article
  • 58 Downloads

Abstract

We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is λ-saturated iff it has cofinality ≥ λ and the underlying order has no (κ, κ)-gaps for regular κ < λ. We also answer a question about balanced pairs of models of PA. Second, assuming instances of GCH, we prove that SOP2 characterizes maximality in the interpretability order ∇*, settling a prior conjecture and proving that SOP2 is a real dividing line. Third, we establish the beginnings of a structure theory for NSOP2, proving that NSOP2 can be characterized by the existence of few so-called higher formulas. In the course of the paper, we show that ps = ts in any weak cofinality spectrum problem closed under exponentiation (naturally defined). We also prove that the local versions of these cardinals need not coincide, even in cofinality spectrum problems arising from Peano arithmetic.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Džamonja and S. Shelah, On ◃*-maximality, Annals of Pure and Applied Logic 125 (2004), 119–158. References are to the extended edition available at http://shelah.logic.at/listb.html, Paper 692.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    H. Gaifman, Models and types of Peano arithmetic, Annals of Mathematical Logic 9 (1976), 223–306.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    H. Gaifman and C. Dimitracopoulos, Fragments of Peano’s arithmetic and the MRDP theorem, in Logic and Algorithmic (Zurich, 1980), Monographies de L’Enseignement Mathématique, Vol. 30, Université de Genève, Geneva, 1982, pp. 187–206.Google Scholar
  4. [4]
    I. Kaplan, S. Shelah and P. Simon, Exact saturation in simple and NIP theories, Journal of Mathematical Logic, accepted.Google Scholar
  5. [5]
    H. J. Keisler, Ultraproducts which are not saturated, Journal of Symbolic Logic 32 (1967), 23–46.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    R. Kossak, private communication.Google Scholar
  7. [7]
    R. Kossak and J. Schmerl, The Structure of Models of Peano Arithmetic, Oxford Logic Guides, Vol. 50, The Clarendon Press, Oxford University Press, Oxford, 2006.CrossRefMATHGoogle Scholar
  8. [8]
    M. Malliaris and S. Shelah, General topology meets model theory,on p and t, Proceedings of the National Academy of Sciences of the United States of America 110 (2013), 13300–13305.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    M. Malliaris and S. Shelah, Cofinality spectrum theorems for model theory, set theory and general topology, Journal of the American Mathematical Society 29 (2016), 237–297.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M. Malliaris and S. Shelah, Existence of optimal ultrafilters and the fundamental complexity of simple theories, Advances in Mathematics 290 (2016), 614–681MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    M. Malliaris and S. Shelah, Keisler’s order has infinitely many classes, Israel Journal of Mathematics, accepted.Google Scholar
  12. [12]
    M. Malliaris and S. Shelah, Open problems on ultrafilters and some connections to the continuum, in Foundations of Mathematics, Contemporary Mathematics, Vol. 690, American Mathematical Society, Providence, RI, to appear.Google Scholar
  13. [13]
    J. Moore, Model theory and the cardinals p and t, Proceedings of the National Academy of Sciences of the United States of America 110 (2013), 13238–13239.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    R. MacDowell and E. Specker, Modelle der arithmetic, in Infinitistic Methods (Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 1959), Pergamon Press, Oxford; Państwowe Wydawnictwo Naukowe, Warsaw, 1961, pp. 257–263.Google Scholar
  15. [15]
    J.-F. Pabion, Saturated models of Peano arithmetic, Journal of Symbolic Logic 47 (1982), 625–637.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    J. Schmerl, Elementary cuts in saturated models of Peano arithmetic, Notre Dame Journal of Formal Logic 53 (2012), 1–13.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    S. Shelah, Classification Theory and the Number of Nonisomorphic Models, Studies in Logic and the Foundations of Mathematics, Vol. 92, North-Holland, Amsterdam–New York, 1978.Google Scholar
  18. [18]
    S. Shelah, End extensions and numbers of countable models, Journal of Symbolic Logic 43 (1978), 550–562.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    S. Shelah, Toward classifying unstable theories, Annals of Pure and Applied Logic 80 (1996), 229–255.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    S. Shelah, General non-structure theory, Paper E59, available at http://shelah.logic.at/Google Scholar
  21. [21]
    S. Shelah, Classification theory for elementary classes with the dependence property—a modest beginning, Scientiae Mathematicae Japonicae 59 (2004), 265–316.MathSciNetMATHGoogle Scholar
  22. [22]
    S. Shelah, Quite complete real closed fields, Israel Journal of Mathematics 142 (2004), 261–272. Extended version available at http://shelah.logic.at/files/757.pdfMathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    S. Shelah, Dependent theories and the generic pair conjecture, Communications in Contemporary Mathematics 17 (2015).Google Scholar
  24. [24]
    S. Shelah, Models of expansions of N with no end extensions, Mathematical Logic Quarterly 57 (2011), 341–365.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    S. Shelah, Atomic saturation of reduced powers, arXiv:1601.04824.Google Scholar
  26. [26]
    S. Shelah and A. Usvyatsov, More on SOP1 and SOP2, Annals of Pure and Applied Logic 155 (2008), 16–31.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    S. Shelah, Dependent dreams: recounting types, arXiv: 1202.5795.Google Scholar
  28. [28]
    A. J. Wilkie and J. B. Paris, On the scheme of induction for bounded arithmetic formulas, Annals of Pure and Applied Logic 35 (1987), 261–302.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Mathematical Sciences Research InstituteBerkeleyUSA
  3. 3.Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat RamThe Hebrew University of JerusalemJerusalemIsrael
  4. 4.Department of Mathematics, Hill Center - Busch Campus RutgersThe State University of New JerseyPiscatawayUSA

Personalised recommendations