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Infinitary stability theory

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Abstract

We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois (orbital) type of length less than a fixed cardinal \(\kappa \). We show:

Theorem 0.1 (The semantic–syntactic correspondence) An AEC K is fully \(({<}\kappa )\)-tame and type short if and only if Galois types are syntactic in the Galois Morleyization.

   This exhibits a correspondence between AECs and the syntactic framework of stability theory inside a model. We use the correspondence to make progress on the stability theory of tame and type short AECs. The main theorems are:

Theorem 0.2 Let K be a \(\text {LS}(K)\)-tame AEC with amalgamation. The following are equivalent:

  1. (1)

    K is Galois stable in some \(\lambda \ge \text {LS}(K)\).

  1. (2)

    K does not have the order property (defined in terms of Galois types).

  1. (3)

    There exist cardinals \(\mu \) and \(\lambda _0\) with \(\mu \le \lambda _0 < \beth _{(2^{\text {LS}(K)})^+}\) such that K is Galois stable in any \(\lambda \ge \lambda _0\) with \(\lambda = \lambda ^{<\mu }\).

Theorem 0.3 Let K be a fully \(({<}\kappa )\)-tame and type short AEC with amalgamation, \(\kappa = \beth _{\kappa } > \text {LS}(K)\). If K is Galois stable, then the class of \(\kappa \)-Galois saturated models of K admits an independence notion (\(({<}\kappa )\)-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory.

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References

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

    Google Scholar 

  2. Boney, W., Grossberg, R.: Forking in short and tame AECs. Preprint arxiv:1306.6562v9

  3. Boney, W., Grossberg, R., Kolesnikov, A., Vasey, S.: Canonical forking in AECs. Preprint arxiv:1404.1494v2

  4. Boney, W., Grossberg, R., Lieberman, M., Rosický, J., Vasey, S.: \(\mu \)-Abstract elementary classes and other generalizations. J. Pure Appl. Algebra. arxiv:1509.07377v2

  5. Baldwin, J.T., Kolesnikov, A.: Categoricity, amalgamation, and tameness. Israel J. Math. 170, 411–443 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boney, W.: Computing the number of types of infinite length. Notre Dame J. Formal Log. arxiv:1309.4485v2

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Boney, W., Vasey, S.: Chains of saturated models in AECs. Preprint arxiv:1503.08781v3

  9. Cherlin, G., Harrington, L., Lachlan, A.H.: \(\aleph _0\)-categorical, \(\aleph _0\)-stable structures. Ann. Pure Appl. Log. 28, 103–135 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dickmann, M.A.: Large infinitary languages. In: Keisler, H.J., Mostowski, A., Robinson, A., Suppes, P., Troelstra, A.S. (eds.) Studies in Logic and the Foundations of Mathematics, vol. 83. North-Holland, Amsterdam (1975)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Grossberg, R.: A course in model theory I, A book in preparation

  13. Grossberg, R.: Indiscernible sequences in a model which fails to have the order property. J. Symb. Log. 56(1), 115–123 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Grossberg, R., Shelah, S.: On Hanf numbers of the infinitary order property, Draft. Preprint arXiv:math/9809196 [math.LO]. Paper number 259 on Shelah’s publication list

  17. Grossberg, R., Shelah, S.: On the number of nonisomorphic models of an infinitary theory which has the infinitary order property, Part A. J. Symb. Log. 51(2), 302–322 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Grossberg, R., VanDieren, M.: Categoricity from one successor cardinal in tame abstract elementary classes. J. Math. Log. 6(2), 181–201 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Grossberg, R., VanDieren, M.: Shelah’s categoricity conjecture from a successor for tame abstract elementary classes. J. Symb. Log. 71(2), 553–568 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hyttinen, T., Kesälä, M.: Independence in finitary abstract elementary classes. Ann. Pure Appl. Log. 143, 103–138 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hyttinen, T., Lessmann, O.: A rank for the class of elementary submodels of a superstable homogeneous model. J. Symb. Log. 67(4), 1469–1482 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hart, B., Shelah, S.: Categoricity over \({P}\) for first order \({T}\) or categoricity for \(\phi \in \mathbb{{L}}_{\omega _1, \omega }\) can stop at \(\aleph _k\) while holding for \(\aleph _0, \ldots, \aleph _{k - 1}\). Israel J. Math. 70, 219–235 (1990)

    Article  MathSciNet  Google Scholar 

  24. Kangas, K.: Finding groups in Zariski-like structures. Preprint arxiv:1404.6811v1

  25. Kirby, J.: On quasiminimal excellent classes. J. Symb. Log. 75(2), 551–564 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kueker, D.W.: Abstract elementary classes and infinitary logics. Ann. Pure Appl. Log. 156, 274–286 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lieberman, M.J.: A topology for Galois types in abstract elementary classes. Math. Log. Q. 57(2), 204–216 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lascar, D., Poizat, B.: An introduction to forking. J. Symb. Log. 44(3), 330–350 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  29. Marcus, L.: A minimal prime model with an infinite set of indiscernibles. Israel J. Math. 11(2), 180–183 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Pillay, A.: Dimension theory and homogeneity for elementary extensions of a model. J. Symb. Log. 47(1), 147–160 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rosický, J.: Concrete categories and infinitary languages. J. Pure Appl. Algebra 22(3), 309–339 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  33. Shelah, S.: Maximal failure of sequence locality. Preprint arxiv:0903.3614v3

  34. Shelah, S.: Finite diagrams stable in power. Ann. Math. Log. 2(1), 69–118 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shelah, S.: A combinatorial problem; stability and order for models and theories in infinitary languages. Pac. J. Math. 41(1), 247–261 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  36. Shelah, S.: Classification theory and the number of non-isomorphic models. In: Keisler, H.J., Mostowski, A., Suppes, P., Troelstra, A.S. (eds.) Studies in Logic and the Foundations of Mathematics, vol. 92. North-Holland, Amsterdam (1978)

    Google Scholar 

  37. 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)

    Google Scholar 

  38. Shelah, S.: Universal classes. In: Baldwin, J.T. (ed.) Classification Theory (Chicago, IL, 1985), Lecture Notes in Mathematics, vol. 1292, pp. 264–418. Springer, Berlin (1987)

    Google Scholar 

  39. Shelah, S.: Classification theory and the number of non-isomorphic models. In: Barwise, J., Keisler, H.J., Suppes, P., Troelstra, A.S. (eds.) Studies in Logic and the Foundations of Mathematics, vol. 92, 2nd edn. North-Holland, Amsterdam (1990)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Shelah, S.: Classification theory for abstract elementary classes. In: Artemov, S., Buss, S., Gabbay, D., Shelah, S., Siekmann, J., van Benthem, J. (eds.) Studies in Logic: Mathematical Logic and Foundations, vol. 18. College Publications (2009)

  42. Shelah, S.: Classification theory for abstract elementary classes 2. In: Artemov, S., Buss, S., Gabbay, D., Shelah, S., Siekmann, J., van Benthem, J. (eds.) Studies in Logic: Mathematical Logic and Foundations, vol. 20. College Publications (2009)

  43. Vasey, S.: Independence in abstract elementary classes. Preprint arxiv:1503.01366v5

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

    Sebastien Vasey

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  1. Sebastien Vasey
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Correspondence to Sebastien Vasey.

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This material is based upon work done while the author was supported by the Swiss National Science Foundation under Grant No. 155136.

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Vasey, S. Infinitary stability theory. Arch. Math. Logic 55, 567–592 (2016). https://doi.org/10.1007/s00153-016-0481-z

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