Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T19:32:12.186Z Has data issue: false hasContentIssue false
  • English
  • Français

EQUIVALENT DEFINITIONS OF SUPERSTABILITY IN TAME ABSTRACT ELEMENTARY CLASSES

Published online by Cambridge University Press:  09 January 2018

RAMI GROSSBERG  and
SEBASTIEN VASEY
RAMI GROSSBERG
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA, USAE-mail:rami@cmu.eduURL: http://math.cmu.edu/∼rami
SEBASTIEN VASEY
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA, USAE-mail:sebv@cmu.eduURL: http://math.cmu.edu/∼svasey/
Get access Rights & Permissions [Opens in a new window]

Abstract

In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent:

  • Corollary.LetKbe a tame AEC with a monster model. Assume thatKis stable in a proper class of cardinals. The following are equivalent:

    1. (1) For all high-enough λ,Khas no long splitting chains.

    2. (2) For all high-enough λ, there exists a good λ-frame on a skeleton ofKλ.

    3. (3) For all high-enough λ,Khas a unique limit model of cardinality λ.

    4. (4) For all high-enough λ,Khas a superlimit model of cardinality λ.

    5. (5) For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated.

    6. (6) There exists μ such that for all high-enough λ,Kis (λ,μ) -solvable.

This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model.

Keywords

Primary 03C48Secondary 03C4503C5203C5503C7503E55abstract elementary classessuperstabilitytamenessindependenceclassification theorysuperlimitsaturatedsolvabilitygood framelimit model
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1387 - 1408
Copyright
Copyright © The Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Albert, M. H. and Grossberg, R., Rich models, this Journal, vol. 55 (1990), no. 3, pp. 1292–1298.Google Scholar
Baldwin, J. T., Categoricity, University Lecture Series, vol. 50, American Mathematical Society, Providence, RI, 2009.Google Scholar
Baldwin, J. T., Grossberg, R., and Shelah, S., Transfering saturation, the finite cover property, and stability, this Journal, vol. 64 (1999), no. 2, pp. 678–684.Google Scholar
Boney, W., Tameness and extending frames. Journal of Mathematical Logic, vol. 14 (2014), no. 2, 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, to appear. Available at http://arxiv.org/abs/1306.6562v11, doi: 10.1016/j.apal.2017.02.002.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., Grossberg, R., VanDieren, M., and Vasey, S., Superstability from categoricity in abstract elementary classes. Annals of Pure and Applied Logic, to appear. Available at http://arxiv.org/abs/1609.07101v3, doi: 10.1016/j.apal.2017.01.005.Google Scholar
Boney, W. and Unger, S., Large cardinal axioms from tameness. Proceedings of the American Mathematical Society, to appear. Available at http://arxiv.org/abs/1509.01191v3, doi: 10.1090/proc/13555.Google Scholar
Boney, W. and Vasey, S., Chains of saturated models in AECs. Archive for Mathematical Logic, to appear. Available at http://arxiv.org/abs/1503.08781v3, doi: 10.1007/s00153-017-0532-0.Google Scholar
Drueck, F., Limit models, superlimit models, and two cardinal problems in abstract elementary classes, Ph.D. thesis, 2013. Available at http://homepages.math.uic.edu/ drueck/thesis.pdf.Google Scholar
Grossberg, R., A downward Löwenheim-Skolem theorem for infinitary theories which have the unsuperstability property, this Journal, vol. 53 (1988), no. 1, pp. 231–242.Google Scholar
Grossberg, R., A Course in Model Theory I, A book in preparation.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 Shelah, S., A nonstructure theorem for an infinitary theory which has the unsuperstability property. Illinois Journal of Mathematics, vol. 30 (1986), no. 2, pp. 364–390.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.Google 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.Google 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
Grossberg, R., VanDieren, M., and Villaveces, A., Uniqueness of limit models in classes with amalgamation. Mathematical Logic Quarterly, vol. 62 (2016), pp. 367–382.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.CrossRefGoogle 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.Google Scholar
Shelah, S., Finite diagrams stable in power. Annals of Mathematical Logic, vol. 2 (1970), no. 1, pp. 69–118.Google Scholar
Shelah, S., Classification of non elementary classes II. Abstract elementary classes, Classification Theory (Chicago, IL, 1985) (Baldwin, J. T., editor), Lecture Notes in Mathematics, vol. 1292, Springer-Verlag, Berlin, Heidelberg, 1987, pp. 419–497.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, 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., Classification Theory for Abstract Elementary Classes 2, Studies in Logic: Mathematical Logic and Foundations, vol. 20, College Publications, London, 2009.Google Scholar
Shelah, S., When first order T has limit models. Colloquium Mathematicum, vol. 126 (2012), pp. 187–204.Google Scholar
Shelah, S., Eventual categoricity spectrum and frames, in preparation (paper number 842 on Shelah’s publication list).Google Scholar
Shelah, S. and Vasey, S., Abstract elementary classes stable in ℵ0, preprint. Available at http://arxiv.org/abs/1702.08281v1.Google Scholar
Shelah, S. and Villaveces, A., Toward categoricity for classes with no maximal models. Annals of Pure and Applied Logic, vol. 97 (1999), pp. 1–25.Google Scholar
VanDieren, M., Categoricity in abstract elementary classes with no maximal models. Annals of Pure and Applied Logic, vol. 141 (2006), pp. 108–147.Google Scholar
VanDieren, M., Erratum to “Categoricity in abstract elementary classes with no maximal models” [Ann. Pure Appl. Logic, 141 (2006), 108–147]. Annals of Pure and Applied Logic, vol. 164 (2013), no. 2, pp. 131–133.Google Scholar
VanDieren, M., Superstability and symmetry. Annals of Pure and Applied Logic, vol. 167 (2016), no. 12, pp. 1171–1183.Google Scholar
VanDieren, M., Symmetry and the union of saturated models in superstable abstract elementary classes. Annals of Pure and Applied Logic, vol. 167 (2016), no. 4, pp. 395–407.Google Scholar
Vasey, S., Building independence relations in abstract elementary classes. Annals of Pure and Applied Logic, vol. 167 (2016), no. 11, pp. 1029–1092.Google Scholar
Vasey, S., Forking and superstability in tame AECs, this Journal, vol. 81 (2016), no. 1, pp. 357–383.Google Scholar
Vasey, S., Infinitary stability theory. Archive for Mathematical Logic, vol. 55 (2016), pp. 567–592.Google Scholar
Vasey, S., Shelah’s eventual categoricity conjecture in universal classes: Part II. Selecta Mathematica, vol. 23 (2017), no. 2, pp. 1469–1506. Available at http://arxiv.org/abs/1602.02633v3.CrossRefGoogle Scholar
Vasey, S., Shelah’s eventual categoricity conjecture in universal classes: part I. Annals of Pure and Applied Logic, to appear. Available at http://arxiv.org/abs/1506.07024v11, doi: 10.1016/j.apal.2017.03.003.Google Scholar
Vasey, S., Saturation and solvability in abstract elementary classes with amalgamation, preprint. Available at http://arxiv.org/abs/1604.07743v2.Google Scholar
Vasey, S., Toward a stability theory of tame abstract elementary classes, preprint. Available at http://arxiv.org/abs/1609.03252v4.Google Scholar
VanDieren, M. and Vasey, S., Symmetry in abstract elementary classes with amalgamation. Archive for Mathematical Logic, to appear. Available at http://arxiv.org/abs/1508.03252v4, doi: 10.1007/s00153-017-0533-z.Google Scholar