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TAMENESS AND FRAMES REVISITED

Published online by Cambridge University Press:  08 September 2017

WILL BONEY  and
SEBASTIEN VASEY
WILL BONEY
Affiliation:
MATHEMATICS DEPARTMENT HARVARD UNIVERSITY CAMBRIDGE, MA02138, USA E-mail: wboney@math.harvard.eduURL: http://math.harvard.edu/∼wboney/
SEBASTIEN VASEY
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA15213, USA E-mail: sebv@cmu.eduURL: http://math.cmu.edu/∼svasey/
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Abstract

We study the problem of extending an abstract independence notion for types of singletons (what Shelah calls a good frame) to longer types. Working in the framework of tame abstract elementary classes, we show that good frames can always be extended to types of independent sequences. As an application, we show that tameness and a good frame imply Shelah’s notion of dimension is well-behaved, complementing previous work of Jarden and Sitton. We also improve a result of the first author on extending a frame to larger models.

Keywords

Primary 03C48Secondary 03C4703C5203C55abstract elementary classesgood framessymmetry
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 3 , September 2017 , pp. 995 - 1021
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

REFERENCES

Baldwin, J. T., Categoricity, University Lecture Series, vol. 50, American Mathematical Society, Providence, RI, 2009.Google Scholar
Baldwin, J. T., Kueker, D., and VanDieren, M., Upward stability transfer for tame abstract elementary classes. Notre Dame Journal of Formal Logic, vol. 47 (2006), no. 2, pp. 291–298.Google Scholar
Boney, W., Computing the number of types of infinite length. Notre Dame Journal of Formal Logic, vol. 48 (2017), no. 1, pp. 133–154.Google Scholar
Boney, W., Tameness and extending frames. Journal of Mathematical Logic, vol. 14 (2014), no. 2, p. 1450007.CrossRefGoogle Scholar
Boney, W., Tameness from large cardinal axioms, this Journal, vol. 79 (2014), no. 4, pp. 1092–1119.Google Scholar
Boney, W. and Grossberg, R., Forking in short and tame AECs. Annals of Pure and Applied Logic, preprint. Available at http://arxiv.org/abs/1306.6562v11.Google Scholar
Boney, W., Grossberg, R., Kolesnikov, A., and Vasey, S., Canonical forking in AECs. Annals of Pure and Applied Logic, vol. 167 (2016), no. 7, pp. 590–613.Google Scholar
Boney, W. and Vasey, S., Good frames in the Hart-Shelah example, preprint. Available at http://arxiv.org/abs/1607.03885v2.Google Scholar
Grossberg, R., Classification theory for abstract elementary classes. Contemporary Mathematics, vol. 302 (2002), pp. 165–204.Google Scholar
Grossberg, R., Iovino, J., and Lessmann, O., A primer of simple theories. Archive for Mathematical Logic, vol. 41 (2002), no. 6, pp. 541–580.Google Scholar
Grossberg, R. and Lessmann, O., Dependence relation in pregeometries. Algebra Universalis, vol. 44 (2000), pp. 199–216.Google Scholar
Grossberg, R. and VanDieren, M., Categoricity from one successor cardinal in tame abstract elementary classes. Journal of Mathematical Logic, vol. 6 (2006), no. 2, pp. 181–201.CrossRefGoogle Scholar
Grossberg, R. and VanDieren, M., Galois-stability for tame abstract elementary classes. Journal of Mathematical Logic, vol. 6 (2006), no. 1, pp. 25–49.CrossRefGoogle Scholar
Grossberg, R. and VanDieren, M., Shelah’s categoricity conjecture from a successor for tame abstract elementary classes, this Journal, vol. 71 (2006), no. 2, pp. 553–568.Google Scholar
Hart, B. and Shelah, S., Categoricity over P for first order T or categoricity for $\phi \in L_{\omega _1 ,\omega } $ can stop at ℵk while holding for ℵ0,...,ℵk–1 . Israel Journal of Mathematics, vol. 70 (1990), pp. 219–235.Google Scholar
Hyttinen, T. and Lessmann, O., 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
Jarden, A., Tameness, uniqueness triples, and amalgamation. Annals of Pure and Applied Logic, vol. 167 (2016), no. 2, pp. 155–188.Google Scholar
Jarden, A. and Shelah, S., Non-forking frames in abstract elementary classes. Annals of Pure and Applied Logic, vol. 164 (2013), pp. 135–191.Google Scholar
Jarden, A. and Sitton, A., Independence, dimension and continuity in non-forking frames, this Journal, vol. 78 (2012), no. 2, pp. 602–632.Google Scholar
Lieberman, M. J., Rank functions and partial stability spectra for tame abstract elementary classes. Notre Dame Journal of Formal Logic, vol. 54 (2013), no. 2, pp. 153–166.CrossRefGoogle Scholar
Makkai, M. and Shelah, S., Categoricity of theories in L k,ω , with κ a compact cardinal. Annals of Pure and Applied Logic, vol. 47 (1990), pp. 41–97.Google Scholar
Shelah, S., Classification Theory and the Number of Non-Isomorphic Models, second ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland, Amsterdam, New York, Oxford, Tokyo, 1990.Google Scholar
Shelah, S., Categoricity for abstract classes with amalgamation. Annals of Pure and Applied Logic, vol. 98 (1999), no. 1, pp. 261–294.Google Scholar
Shelah, S., Classification Theory for Abstract Elementary Classes, Studies in Logic: Mathematical Logic and Foundations, vol. 18, College Publications, London, 2009.Google Scholar
Shelah, S., Building independence relations in abstract elementary classes. Annals of Pure and Applied Logic, vol. 167 (2016), no. 11, pp. 1029–1092.Google Scholar
Shelah, S., Forking and superstability in tame AECs, this Journal, vol. 81 (2016), no. 1, pp. 357–383.Google Scholar
Shelah, S., Downward categoricity from a successor inside a good frame. Annals of Pure and Applied Logic, vol. 168 (2017), no. 3, pp. 651–692.Google Scholar
Vasey, S., Quasiminimal abstract elementary classes, preprint. Available at https://arxiv.org/abs/1611.07380v3.Google Scholar
Vasey, S., Shelah’s eventual categoricity conjecture in universal classes: Part I, preprint. Available at http://arxiv.org/abs/1506.07024v10.Google Scholar
Vasey, S., Shelah’s eventual categoricity conjecture in universal classes. Part II. Selecta Mathematica, to appear. Available at http://arxiv.org/abs/1602.02633v2.Google Scholar