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)
K is Galois stable in some \(\lambda \ge \text {LS}(K)\).
-
(2)
K does not have the order property (defined in terms of Galois types).
-
(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
Baldwin, J.T.: Categoricity, University Lecture Series, vol. 50. American Mathematical Society, Providence (2009)
Boney, W., Grossberg, R.: Forking in short and tame AECs. Preprint arxiv:1306.6562v9
Boney, W., Grossberg, R., Kolesnikov, A., Vasey, S.: Canonical forking in AECs. Preprint arxiv:1404.1494v2
Boney, W., Grossberg, R., Lieberman, M., Rosický, J., Vasey, S.: \(\mu \)-Abstract elementary classes and other generalizations. J. Pure Appl. Algebra. arxiv:1509.07377v2
Baldwin, J.T., Kolesnikov, A.: Categoricity, amalgamation, and tameness. Israel J. Math. 170, 411–443 (2009)
Boney, W.: Computing the number of types of infinite length. Notre Dame J. Formal Log. arxiv:1309.4485v2
Boney, W.: Tameness from large cardinal axioms. J. Symb. Log. 79(4), 1092–1119 (2014)
Boney, W., Vasey, S.: Chains of saturated models in AECs. Preprint arxiv:1503.08781v3
Cherlin, G., Harrington, L., Lachlan, A.H.: \(\aleph _0\)-categorical, \(\aleph _0\)-stable structures. Ann. Pure Appl. Log. 28, 103–135 (1985)
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)
Grossberg, R., Lessmann, O.: Shelah’s stability spectrum and homogeneity spectrum in finite diagrams. Arch. Math. Logic 41(1), 1–31 (2002)
Grossberg, R.: A course in model theory I, A book in preparation
Grossberg, R.: Indiscernible sequences in a model which fails to have the order property. J. Symb. Log. 56(1), 115–123 (1991)
Grossberg, R.: On chains of relatively saturated submodels of a model without the order property. J. Symb. Log. 56, 124–128 (1991)
Grossberg, R.: Classification theory for abstract elementary classes. Contemp. Math. 302, 165–204 (2002)
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
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)
Grossberg, R., VanDieren, M.: Categoricity from one successor cardinal in tame abstract elementary classes. J. Math. Log. 6(2), 181–201 (2006)
Grossberg, R., VanDieren, M.: Galois-stability for tame abstract elementary classes. J. Math. Log. 6(1), 25–49 (2006)
Grossberg, R., VanDieren, M.: Shelah’s categoricity conjecture from a successor for tame abstract elementary classes. J. Symb. Log. 71(2), 553–568 (2006)
Hyttinen, T., Kesälä, M.: Independence in finitary abstract elementary classes. Ann. Pure Appl. Log. 143, 103–138 (2006)
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)
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)
Kangas, K.: Finding groups in Zariski-like structures. Preprint arxiv:1404.6811v1
Kirby, J.: On quasiminimal excellent classes. J. Symb. Log. 75(2), 551–564 (2010)
Kueker, D.W.: Abstract elementary classes and infinitary logics. Ann. Pure Appl. Log. 156, 274–286 (2008)
Lieberman, M.J.: A topology for Galois types in abstract elementary classes. Math. Log. Q. 57(2), 204–216 (2011)
Lascar, D., Poizat, B.: An introduction to forking. J. Symb. Log. 44(3), 330–350 (1979)
Marcus, L.: A minimal prime model with an infinite set of indiscernibles. Israel J. Math. 11(2), 180–183 (1972)
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)
Pillay, A.: Dimension theory and homogeneity for elementary extensions of a model. J. Symb. Log. 47(1), 147–160 (1982)
Rosický, J.: Concrete categories and infinitary languages. J. Pure Appl. Algebra 22(3), 309–339 (1981)
Shelah, S.: Maximal failure of sequence locality. Preprint arxiv:0903.3614v3
Shelah, S.: Finite diagrams stable in power. Ann. Math. Log. 2(1), 69–118 (1970)
Shelah, S.: A combinatorial problem; stability and order for models and theories in infinitary languages. Pac. J. Math. 41(1), 247–261 (1972)
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)
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)
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)
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)
Shelah, S.: Categoricity for abstract classes with amalgamation. Ann. Pure Appl. Log. 98(1), 261–294 (1999)
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)
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)
Vasey, S.: Independence in abstract elementary classes. Preprint arxiv:1503.01366v5
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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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DOI: https://doi.org/10.1007/s00153-016-0481-z