## Notre Dame Journal of Formal Logic

### Comparing Borel Reducibility and Depth of an ω-Stable Theory

Martin Koerwien

#### Abstract

In "A proof of Vaught's conjecture for ω-stable theories," the notions of ENI-NDOP and eni-depth have been introduced, which are variants of the notions of NDOP and depth known from Shelah's classification theory. First, we show that for an ω-stable first-order complete theory, ENI-NDOP allows tree decompositions of countable models. Then we discuss the relationship between eni-depth and the complexity of the isomorphism relation for countable models of such a theory in terms of Borel reducibility as introduced by Friedman and Stanley and construct, in particular, a sequence of complete first-order ω-stable theories $(T_\alpha)_{\alpha < \omega_1}$ with increasing and cofinal eni-depth and isomorphism relations which are strictly increasing with respect to Borel reducibility.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 50, Number 4 (2009), 365-380.

Dates
First available in Project Euclid: 11 February 2010

https://projecteuclid.org/euclid.ndjfl/1265899120

Digital Object Identifier
doi:10.1215/00294527-2009-016

Mathematical Reviews number (MathSciNet)
MR2598869

Zentralblatt MATH identifier
1203.03045

Subjects
Primary: 03C15 03C45 03E15

#### Citation

Koerwien , Martin. Comparing Borel Reducibility and Depth of an ω-Stable Theory. Notre Dame J. Formal Logic 50 (2009), no. 4, 365--380. doi:10.1215/00294527-2009-016. https://projecteuclid.org/euclid.ndjfl/1265899120

#### References

• [1] Baldwin, J. T., Fundamentals of Stability Theory, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1988.
• [2] Becker, H., and A. S. Kechris, The Descriptive Set Theory of Polish Group Actions, vol. 232 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1996.
• [3] Bouscaren, E., "Martin's conjecture for $\omega$-stable theories", Israel Journal of Mathematics, vol. 49 (1984), pp. 15--25.
• [4] Bouscaren, E., and D. Lascar, "Countable models of nonmultidimensional $\aleph \sb{0}$-stable theories", The Journal of Symbolic Logic, vol. 48 (1983), pp. 197--205.
• [5] Friedman, H., and L. Stanley, "A Borel reducibility theory for classes of countable structures", The Journal of Symbolic Logic, vol. 54 (1989), pp. 894--914.
• [6] Harrington, L. A., A. S. Kechris, and A. Louveau, "A Glimm-Effros dichotomy for Borel equivalence relations", Journal of the American Mathematical Society, vol. 3 (1990), pp. 903--28.
• [7] Hjorth, G., "Countable models and the theory of Borel equivalence relations", pp. 1--43 in The Notre Dame Lectures, vol. 18 of Lecture Notes in Logic, Association for Symbolic Logic, Urbana, 2005.
• [8] Hjorth, G., and A. S. Kechris, "Borel equivalence relations and classifications of countable models", Annals of Pure and Applied Logic, vol. 82 (1996), pp. 221--72.
• [9] Hjorth, G., and A. S. Kechris, "New dichotomies for Borel equivalence relations", The Bulletin of Symbolic Logic, vol. 3 (1997), pp. 329--46.
• [10] Hjorth, G., A. S. Kechris, and A. Louveau, "Borel equivalence relations induced by actions of the symmetric group", Annals of Pure and Applied Logic, vol. 92 (1998), pp. 63--112.
• [11] Jackson, S., A. S. Kechris, and A. Louveau, "Countable Borel equivalence relations", Journal of Mathematical Logic, vol. 2 (2002), pp. 1--80.
• [12] Kechris, A. S., Classical Descriptive Set Theory, vol. 156 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1995.
• [13] Koerwien, M., La complexité de la relation d'isomorphisme pour les modèles dénombrables des théories $\omega$-stables, Ph.D. thesis, Université Paris 7, 2007.
• [14] Laskowski, M. C., and S. Shelah, "Borel completeness of some $\omega$-stable theories". forthcoming.
• [15] Laskowski, M. C., "An old friend revisited: Countable models of $\omega$-stable theories", Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 133--41.
• [16] Louveau, A., and B. Veličković, "A note on Borel equivalence relations", Proceedings of the American Mathematical Society, vol. 120 (1994), pp. 255--59.
• [17] Marker, D., "The Borel complexity of isomorphism for theories with many types", Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 93--97.
• [18] Pillay, A., "Weakly homogeneous models", Proceedings of the American Mathematical Society, vol. 86 (1982), pp. 126--32.
• [19] Shelah, S., L. Harrington, and M. Makkai, "A proof of Vaught's conjecture for $\omega$-stable theories", Israel Journal of Mathematics, vol. 49 (1984), pp. 259--80.
• [20] Shelah, S., Classification Theory and the Number of Nonisomorphic Models, vol. 92 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1978.