Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T03:34:09.727Z Has data issue: false hasContentIssue false
  • English
  • Français

A complicated ω-stable depth 2 theory

Published online by Cambridge University Press:  12 March 2014

Martin Koerwien
Martin Koerwien*
Affiliation:
Centre de Recerca Matematica, 08193 Bellaterra, Barcelona, Spain, E-mail: koerwien@math.uic.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a countable complete first order theory T which is model theoretically very well behaved: it eliminates quantifiers, is ω-stable, it has NDOP and is shallow of depth two. On the other hand, there is no countable bound on the Scott heights of its countable models, which implies that the isomorphism relation for countable models is not Borel.

Keywords

03C1503C4503E15omega-stabilityclassificationcountable modelsBorel reducibilityScott height
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 1 , March 2011 , pp. 47 - 65
Copyright
Copyright © Association for Symbolic Logic 2011

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

[1]Baldwin, J. T., Fundamentals of stability theory, Springer, 1988.CrossRefGoogle Scholar
[2]Becker, H. and Kechris, A. S., The descriptive set theory of polish group actions, London Mathematical Society Lecture Notes Series 232, Cambridge University Press, 1996.CrossRefGoogle Scholar
[3]Harrington, L. A., Kechris, A. S., and Louveau, A., A Glimm–Effros dichotomy for Borel equivalence relations, Journal of the American Mathematical Society, vol. 3 (1990), pp. 663–693.CrossRefGoogle Scholar
[4]Hjorth, G., Countable models and the theory of Borel equivalence relations, Notre Dame Lecture Notes in Logic, vol. 18 (2005), pp. 1–43.Google Scholar
[5]Hjorth, G. and Kechris, A. S., Borel equivalence relations and classifications of countable models, Annals of Pure and Applied Logic, vol. 82 (1996), pp. 221–272.CrossRefGoogle Scholar
[6]Hjorth, G., Kechris, A. S., and Louveau, A., Borel equivalence relations induced by actions of the symmetric group, Annals of Pure and Applied Logic, vol. 92 (1998), pp. 63–112.CrossRefGoogle Scholar
[7]Hjorth, Greg and Kechris, Alexander, New dichotomies for borel equivalence relations, The Bulletin of Symbolic Logic, vol. 3 (1997), no. 3, pp. 329–346.CrossRefGoogle Scholar
[8]Jackson, S., Kechris, A. S., and Louveau, A., Countable Borel equivalence relations, Journal of Mathematical Logic, vol. 2 (2002), no. 1, pp. 1–80.CrossRefGoogle Scholar
[9]Koerwien, M., La complexité de la relation d'isomorphisme pour les modèles dènombrables d'une théorie oméga-stable, Ph.D. thesis, Université Paris 7, 2007, available at www.math.uic.edu/~koerwien.Google Scholar
[10]Koerwien, M., Comparing borel reducibility and depth of an ω-stable theory, to appear.Google Scholar
[11]Lascar, D., Why some people are excited by vaught's conjecture, this Journal, vol. 50 (1985), pp. 973–982.Google Scholar
[12]Louveau, A. and Velickovic, B., A note on Borel equivalence relations, Proceedings of the American Mathematical Society, vol. 120 (1994), pp. 255–259.CrossRefGoogle Scholar
[13]Shelah, S., Classification theory and the number of nonisomorphic models, North Holland, 1978.Google Scholar