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Chains of saturated models in AECs

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Abstract

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove:

Theorem 0.1. If K is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough \(\lambda {:}\)

  1. (1)

    The union of an increasing chain of \(\lambda \)-saturated models is \(\lambda \)-saturated.

  2. (2)

    There exists a type-full good \(\lambda \) -frame with underlying class the saturated models of size \(\lambda \).

  3. (3)

    There exists a unique limit model of size \(\lambda \).

Our proofs use independence calculus and a generalization of averages to this non first-order context.

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References

  1. Albert, M.H., Grossberg, R.: Rich models. J. Symb. Log. 55(3), 1292–1298 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baldwin, J.T.: Categoricity, University Lecture Series, vol. 50, American Mathematical Society (2009)

  3. Boney, W., Grossberg, R.: Forking in short and tame AECs. Ann. Pure Appl. Log. (2017). doi:10.1016/j.apal.2017.02.002

  4. Boney, W., Grossberg, R., Kolesnikov, A., Vasey, S.: Canonical forking in AECs. Ann. Pure Appl. Log. 167(7), 590–613 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boney, W.: Tameness and extending frames. J. Math. Log. 14(2), 1450007 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boney, W.: Tameness from large cardinal axioms. J. Symb. Log. 79(4), 1092–1119 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Drueck, F.: Limit models, superlimit models, and two cardinal problems in abstract elementary classes, Ph.D. thesis (2013). http://homepages.math.uic.edu/~drueck/thesis

  8. Grossberg, R., Lessmann, O.: Shelah’s stability spectrum and homogeneity spectrum in finite diagrams. Arch. Math. Log. 41(1), 1–31 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grossberg, R.: A Course in Model Theory I (in preparation)

  10. Grossberg, R.: On chains of relatively saturated submodels of a model without the order property. J. Symb. Log. 56, 124–128 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grossberg, R.: Classification theory for abstract elementary classes. Contemp. Math. 302, 165–204 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grossberg, R., VanDieren, M.: Galois-stability for tame abstract elementary classes. J. Math. Log. 6(1), 25–49 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grossberg, R., VanDieren, M., Villaveces, A.: Uniqueness of limit models in classes with amalgamation. Math. Log. Q. 62, 367–382 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Harnik, V.: On the existence of saturated models of stable theories. Proc. Am. Math. Soc. 52, 361–367 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  15. Makkai, M., Shelah, S.: Categoricity of theories in \({L}_{\kappa,\omega }\), with \(\kappa \) a compact cardinal. Ann. Pure Appl. Log. 47, 41–97 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shelah, S.: The lazy model theoretician’s guide to stability. Log. Anal. 18, 241–308 (1975)

    MathSciNet  MATH  Google Scholar 

  17. Shelah, S.: Classification of non elementary classes II. Abstract elementary classes. In: Baldwin, J.T. (ed.) Classification Theory (Chicago, IL, 1985), Lecture Notes in Mathematics, vol. 1292, pp. 419–497. Springer, Berlin (1987)

  18. Shelah, S.: Classification theory and the number of non-isomorphic models. In: Studies in Logic and the Foundations of Mathematics, vol. 92, 2nd edn. North-Holland, Amsterdam (1990)

  19. Shelah, S.: Categoricity for abstract classes with amalgamation. Ann. Pure Appl. Log. 98(1), 261–294 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shelah, S.: Classification Theory for Abstract Elementary Elasses, Studies in Logic: Mathematical Logic and Foundations, vol. 18. College Publications, Norcross (2009)

    Google Scholar 

  21. Shelah, S.: Classification Theory for Abstract Elementary Classes 2, Studies in Logic: Mathematical Logic and Foundations, vol. 20. College Publications, Norcross (2009)

    Google Scholar 

  22. Shelah, S., Villaveces, A.: Toward categoricity for classes with no maximal models. Ann. Pure Appl. Log. 97, 1–25 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. VanDieren, M.: Categoricity in abstract elementary classes with no maximal models. Ann. Pure Appl. Log. 141, 108–147 (2006)

    Article  MathSciNet  Google Scholar 

  24. VanDieren, M.: Erratum to “Categoricity in abstract elementary classes with no maximal models” [Ann. Pure Appl. Logic 141 (2006) 108–147]. Ann. Pure Appl. Log. 164(2), 131–133 (2013)

    Article  MathSciNet  Google Scholar 

  25. Vasey, S.: Building independence relations in abstract elementary classes. Ann. Pure Appl. Log. 167(11), 1029–1092 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Vasey, S.: Forking and superstability in tame AECs. J. Symb. Log. 81(1), 357–383 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Vasey, S.: Infinitary stability theory. Arch. Math. Log. 55, 567–592 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, USA

    Will Boney

  2. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA

    Sebastien Vasey

Authors
  1. Will Boney
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  2. Sebastien Vasey
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Correspondence to Sebastien Vasey.

Additional information

This material is based upon work done while Will Boney was supported by the National Science Foundation under Grant No. DMS-1402191 and Sebastien Vasey was supported by the Swiss National Science Foundation under Grant No. 155136.

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Boney, W., Vasey, S. Chains of saturated models in AECs. Arch. Math. Logic 56, 187–213 (2017). https://doi.org/10.1007/s00153-017-0532-0

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  • DOI: https://doi.org/10.1007/s00153-017-0532-0

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