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Categoricity transfer in simple finitary abstract elementary classes

Published online by Cambridge University Press:  12 March 2014

Tapani Hyttinen  and
Meeri Kesälä
Tapani Hyttinen
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin Katu 2B), FI-00014 University of Helsinki, Finland, E-mail: tapani.hyttinen@helsinki.fi, E-mail: meeri.kesala@helsinki.fi
Meeri Kesälä
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Gustaf Hällströmin Katu 2B), FI-00014 University of Helsinki, Finland, E-mail: tapani.hyttinen@helsinki.fi, E-mail: meeri.kesala@helsinki.fi
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Abstract

We continue our study of finitary abstract elementary classes, defined in [7]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak κ-categoricity and f-primary models to the framework of ℵ0-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let () be a simple finitary AEC, weakly categorical in some uncountable κ. Then () is weakly categorical in each λ ≥ min. If the class () is also -tame, weak κ-categoricity is equivalent with κ-categoricity in the usual sense.

We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 3 , September 2011 , pp. 759 - 806
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

[1]Baldwin, John T., Categoricity, University Lecture Series, vol. 50, American Mathematical Society, 2009.Google Scholar
[2]Baldwin, John T. and Kolesnikov, Alexei, Categoricity, amalgamation, and lameness, Israel Journal of Mathematics, vol. 170 (2009), pp. 411–443.CrossRefGoogle Scholar
[3]Grossberg, Rami and VanDieren, Monica, Categoricity from one successor cardinal in tame abstract elementary classes, Journal of Mathematical Logic, vol. 6 (2006), no. 2, pp. 181–201.CrossRefGoogle Scholar
[4]Grossberg, Rami and VanDieren, Monica, Galois-stability in tame abstract elementary classes, Journal of Mathematical Logic, vol. 6 (2006), no. 1, pp. 25–49.CrossRefGoogle Scholar
[5]Grossberg, Rami and VanDieren, Monica, Shelah's categoricity conjecture from a successor for tame abstract elementary classes, this Journal, vol. 71 (2006), no. 2, pp. 553–568.Google Scholar
[6]Hart, Bradd and Shelah, Saharon, Categoricity over P for first order T or categoricity for ψ ∈ 1ω 1ω can stop at ℵk while holding for ℵ0,…, ℵk−1, Israel Journal of Mathematics, vol. 70 (1990), pp. 219–235.Google Scholar
[7]Hyttinen, T. and Kesälä, M., Independence in finitary abstract elementary classes, Annals of Pure and Applied Logic, vol. 143 (2006), no. 1–3, pp. 103–138.CrossRefGoogle Scholar
[8]Hyttinen, Tapani and Kesälä, Meeri, Finitary abstract elementary classes, Ph.D. thesis, University of Helsinki, Department of Mathematics and Statistics, 2006.CrossRefGoogle Scholar
[9]Hyttinen, Tapani and Kesälä, Meeri, Superstability in simple finitary AEC, Fundamenta Mathematicae, vol. 195 (2007), no. 3, pp. 221–268.CrossRefGoogle Scholar
[10]Hyttinen, Tapani and Lessmann, Olivier, A rank for the class of elementary submodels of a superstable homogeneous model, this Journal, vol. 67 (2002), no. 4, pp. 1469–1482.Google Scholar
[11]Hyttinen, Tapani and Lessmann, Olivier, Simplicity and uncountable categoricity in excellent classes, Annals of Pure and Applied Logic, vol. 139 (2006), no. 1–3, pp. 110–137.CrossRefGoogle Scholar
[12]Hyttinen, Tapani, Lessmann, Olivier, and Shelah, Saharon, Interpreting groups and fields in some nonelementary classes, Journal of Mathematical Logic, vol. 5 (2005), no. 1, pp. 1–47.CrossRefGoogle Scholar
[13]Keisler, H. Jerome, Model theory for infinitary logic, Studies in Logic and the Foundations of Mathematics, North-Holland, 1971.Google Scholar
[14]Kueker, David W., Abstract elementary classes and infinitary logics, Annals of Pure and Applied Logic, vol. 156 (2008), no. 2–3, pp. 274–286.CrossRefGoogle Scholar
[15]Laskowski, Michael C., An old friend revisited: countable models of ω-stable theories, Notre Dame Journal of Formal Logic, vol. 48 (2007), no. 1, pp. 133–141.CrossRefGoogle Scholar
[16]Lessmann, Olivier, An introduction to excellent classes, Logic and its applications (Blass, Andreas and Zhang, Yi, editors), Contemporary Mathematics, vol. 380, American Mathematical Society, 2005, pp. 231–259.CrossRefGoogle Scholar
[17]Lessmann, Olivier, An introduction to uncountable categoricity in abstract elementary classes, Graduate Texts in Mathematics, vol. 6, University of Helsinki, Department of Mathematics and Statistics, 2005.Google Scholar
[18]Lessmann, Olivier, Upward categoricity from a successor cardinal for tame abstract classes with amalgamation, this Journal, vol. 70 (2005), no. 2, pp. 639–660.Google Scholar
[19]Shelah, Saharon, Finite diagrams stable in power, Annals of Mathematical Logic, vol. 2 (1970), pp. 293–300.CrossRefGoogle Scholar
[20]Shelah, Saharon, Classification theory and the number of nonisomorphic models, North-Holland, 1978, second revised edition, 1990.Google Scholar
[21]Shelah, Saharon, Classification theory for for nonelementary classes, I. The number of uncountable models of ψ ∈ Lω 1,ω, Parts A and B, Israel Journal of Mathematics, vol. 46 (1983), pp. 212–273.CrossRefGoogle Scholar
[22]Shelah, Saharon, Classification of non elementary classes II, Abstract elementary classes, Classification theory: Proceedings of the V.S.-Israel Workshop on Model Theory in Mathematical Logic, Chicago, 1985 (Baldwin, John T., editor), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, Berlin, 1987, pp. 419–497.CrossRefGoogle Scholar
[23]Shelah, Saharon, Categoricity of abstract classes with amalgamation, Annals of Pure and Applied Logic, vol. 98 (1999), pp. 261–294.CrossRefGoogle Scholar
[24]Viljanen, Meeri, Independence in local abstract elementary classes, Licantiate's thesis, University of Helsinki, 2005.Google Scholar