## Notre Dame Journal of Formal Logic

### An Old Friend Revisited: Countable Models of ω-Stable Theories

#### Abstract

We work in the context of ω-stable theories. We obtain a natural, algebraic equivalent of ENI-NDOP and discuss recent joint proofs with Shelah that if an ω-stable theory has either ENI-DOP or is ENI-NDOP and is ENI-deep, then the set of models of T with universe ω is Borel complete.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 48, Number 1 (2007), 133-141.

Dates
First available in Project Euclid: 1 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1172787550

Digital Object Identifier
doi:10.1305/ndjfl/1172787550

Mathematical Reviews number (MathSciNet)
MR2289902

Zentralblatt MATH identifier
1130.03026

Subjects
Primary: 03C45: Classification theory, stability and related concepts [See also 03C48]
Secondary: 04A15

#### Citation

Laskowski, Michael C. An Old Friend Revisited: Countable Models of ω-Stable Theories. Notre Dame J. Formal Logic 48 (2007), no. 1, 133--141. doi:10.1305/ndjfl/1172787550. https://projecteuclid.org/euclid.ndjfl/1172787550

#### References

• [1] Becker, H., and A. S. Kechris, Borel actions of Polish groups,'' American Mathematical Society. Bull. New Ser., vol. 28 (1993), pp. 334--41.
• [2] Bouscaren, E., and D. Lascar, "Countable models of nonmultidimensional $\aleph \sb0$"-stable theories, The Journal of Symbolic Logic, vol. 48 (1983), pp. 197--205.
• [3] 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.
• [4] Harrington, L., and M. Makkai, "An exposition of Shelah's `main gap': Counting uncountable models of $\omega$"-stable and superstable theories, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 139--77.
• [5] Hjorth, G., and A. S. Kechris, "Recent developments in the theory of Borel reducibility", Fundamenta Mathematicae, vol. 170 (2001), pp. 21--52. Dedicated to the memory of Jerzy Łoś.
• [6] Laskowski, M. C., and S. Shelah, "Borel completeness of some $\aleph_0$-stable theories", in preparation.
• [7] Marker, D., "The number of countable differentially closed fields", Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 99--113 (electronic).
• [8] Shelah, S., Classification Theory and the Number of Nonisomorphic Models, 2d edition, vol. 92 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990.
• [9] 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.
• [10] Shelah, S., "Characterizing an $\aleph\sb \epsilon$"-saturated model of superstable NDOP theories by its $\Bbb L\sb\infty,\aleph\sb \epsilon$-theory, Israel Journal of Mathematics, vol. 140 (2004), pp. 61--111.
• [11] Vaught, R., "Invariant sets in topology and logic", Fundamenta Mathematicae, vol. 82 (1974/75), pp. 269--94. Collection of articles dedicated to Andrzej Mostowski on his 60th birthday, VII.