Advertisement

Archive for Mathematical Logic

, Volume 57, Issue 5–6, pp 687–712 | Cite as

Good frames in the Hart–Shelah example

  • Will BoneyEmail author
  • Sebastien Vasey
Article

Abstract

For a fixed natural number \(n \ge 1\), the Hart–Shelah example is an abstract elementary class (AEC) with amalgamation that is categorical exactly in the infinite cardinals less than or equal to \(\aleph _{n}\). We investigate recently-isolated properties of AECs in the setting of this example. We isolate the exact amount of type-shortness holding in the example and show that it has a type-full good \(\aleph _{n - 1}\)-frame which fails the existence property for uniqueness triples. This gives the first example of such a frame. Along the way, we develop new tools to build and analyze good frames.

Keywords

Abstract elementary classes Hart–Shelah example Good frames Uniqueness triples 

Mathematics Subject Classification

Primary 03C48 Secondary 03C45 03C52 03C55 03C75 03E55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baldwin, J.T.: Categoricity. University Lecture Series, vol. 50. American Mathematical Society (2009)Google Scholar
  2. 2.
    Boney, W., Grossberg, R., Kolesnikov, A., Vasey, S.: Canonical forking in AECs. Ann. Pure Appl. Log. 167(7), 590–613 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boney, W., Grossberg, R., VanDieren, M., Vasey, S.: Superstability from categoricity in abstract elementary classes. Ann. Pure Appl. Log.  https://doi.org/10.1016/j.apal.2017.01.005. http://arxiv.org/abs/1609.07101v3 (to appear)
  4. 4.
    Baldwin, J.T., Kolesnikov, A.: Categoricity, amalgamation, and tameness. Isr. J. Math. 170, 411–443 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boney, W.: Tameness and extending frames. J. Math. Log. 14(2), 1450007 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Boney, W.: Tameness from large cardinal axioms. J. Symb. Log. 79(4), 1092–1119 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boney, W., Vasey, S.: Chains of saturated models in AECs. Arch. Math. Log. http://arxiv.org/abs/1503.08781v4 (to appear)
  8. 8.
    Boney, W., Vasey, S.: Tameness and frames revisited. J. Symb. Log. http://arxiv.org/abs/1406.5980v6 (to appear)
  9. 9.
    Grossberg, R., Vasey, S.: Equivalent definitions of superstability in tame abstract elementary classes. J. Symb. Log. http://arxiv.org/abs/1507.04223v5 (to appear)
  10. 10.
    Grossberg, R., VanDieren, M.: Galois-stability for tame abstract elementary classes. J. Math. Log. 6(1), 25–49 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grossberg, R., VanDieren, M., Villaveces, A.: Uniqueness of limit models in classes with amalgamation. Math. Log. Q. 62, 367–382 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hart, B., Shelah, S.: Categoricity over \({P}\) for first order \({T}\) or categoricity for \(\phi \in {L}_{\omega _1, \omega }\) can stop at \(\aleph _k\) while holding for \(\aleph _0, \ldots, \aleph _{k - 1}\). Isr. J. Math. 70, 219–235 (1990)CrossRefGoogle Scholar
  13. 13.
    Jarden, A.: Tameness, uniqueness triples, and amalgamation. Ann. Pure Appl. Log. 167(2), 155–188 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jarden, A., Shelah, S.: Non-forking frames in abstract elementary classes. Ann. Pure Appl. Log. 164, 135–191 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shelah, S.: Categoricity in \(\aleph _1\) of sentences in \({L}_{\omega _1, \omega }({Q})\). Isr. J. Math. 20(2), 127–148 (1975)CrossRefGoogle Scholar
  16. 16.
    Shelah, S.: Classification theory for non-elementary classes I: the number of uncountable models of \(\psi \in {L}_{\omega _1, \omega }\). Part A. Isr. J. Math. 46(3), 214–240 (1983)CrossRefGoogle Scholar
  17. 17.
    Shelah, S.: Categoricity for abstract classes with amalgamation. Ann. Pure Appl. Log. 98(1), 261–294 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shelah, S.: Classification theory for abstract elementary classes. In: Gabbay, D.M. (eds.) Studies in Logic: Mathematical Logic and Foundations, vol. 18. College Publications, IL (2009)Google Scholar
  19. 19.
    Shelah, S.: Classification theory for abstract elementary classes 2. In: Gabbay, D.M. (eds.) Studies in Logic: Mathematical Logic and Foundations, vol. 20. College Publications, IL (2009)Google Scholar
  20. 20.
    Shelah, S., Villaveces, A.: Toward categoricity for classes with no maximal models. Ann. Pure Appl. Log. 97, 1–25 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    VanDieren, M.: Categoricity in abstract elementary classes with no maximal models. Ann. Pure Appl. Log. 141, 108–147 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    VanDieren, M.: Superstability and symmetry. Ann. Pure Appl. Log. 167(12), 1171–1183 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vasey, S.: Quasiminimal abstract elementary classes. https://arxiv.org/abs/1611.07380v3 (preprint)
  24. 24.
    Vasey, S.: Saturation and solvability in abstract elementary classes with amalgamation. http://arxiv.org/abs/1604.07743v2 (preprint)
  25. 25.
    Vasey, S.: Building independence relations in abstract elementary classes. Ann. Pure Appl. Log. 167(11), 1029–1092 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vasey, S.: Forking and superstability in tame AECs. J. Symb. Log. 81(1), 357–383 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Vasey, S.: Infinitary stability theory. Arch. Math. Log. 55, 567–592 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Vasey, S.: Downward categoricity from a successor inside a good frame. Ann. Pure Appl. Log. 168(3), 651–692 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    VanDieren, M., Vasey, S.: Symmetry in abstract elementary classes with amalgamation. Arch. Math. Log.  https://doi.org/10.1007/s00153-017-0533-z. http://arxiv.org/abs/1508.03252v4 (to appear)

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations