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On atomic or saturated sets

Published online by Cambridge University Press:  12 March 2014

Ludomir Newelski
Ludomir Newelski*
Affiliation:
Impan, ul. Kopernika 18, 51-617 Wrocław, Poland, E-mail: newelski@math.uni.wroc.pl
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Abstract

Assume T is stable, small and Φ(x) is a formula of L(T). We study the impact on T⌈Φ of naming finitely many elements of a model of T. We consider the cases of T⌈Φ which is ω-stable or superstable of finite rank. In these cases we prove that if T has countable models and Q = Φ(M) is countable and atomic or saturated, then any good type in S(Q) is τ-stable. If T⌈Φ is ω-stable and (bounded, 1-based or of finite rank) with , then we prove that every good p ∈ S(Q) is τ-stable for any countable Q. The proofs of these results lead to several new properties of small stable theories, particularly of types of finite weight in such theories.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 1 , March 1996 , pp. 318 - 333
Copyright
Copyright © Association for Symbolic Logic 1996

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References

REFERENCES

[1]Baldwin, J. T., Fundamentals of stability theory, Springer, 1987.Google Scholar
[2]Bouscaren, E., Countable models of multidimensional ℵ0-stable theories, this Journal, vol. 48 (1983), pp. 377–383.Google Scholar
[3]Buechler, S., Vaught's conjecture for superstable theories of finite rank, preprint, 10 1993.Google Scholar
[4]Lachlan, A., Two conjectures regarding the stability of ω-categorical theories, Fundamenta Mathematicae, vol. 81 (1974), pp. 133–145.CrossRefGoogle Scholar
[5]Newelski, L., A model and its subset, this Journal, vol. 57 (1992), pp. 644–658.Google Scholar
[6]Newelski, L., Scott analysis of pseudo-types, this Journal, vol. 58 (1993), pp. 648–663.Google Scholar
[7]Newelski, L., Meager forking, Ann. Pure Appl. Logic, vol. 70 (1994), pp. 141–175.CrossRefGoogle Scholar
[8]Pillay, A., Forking, normalization and canonical bases, Annals of Pure and Applied Logic, vol. 32 (1986), pp. 61–81.CrossRefGoogle Scholar
[9]Shelah, S., Classification theory, second ed., Springer, North Holland, 1990.Google Scholar
[10]Shelah, S., Harrington, L., and Makkai, M., A proof of Vaught's conjecture for ω-stable theories, Israel Journal of Mathematics, vol. 49 (1984), pp. 259–280.CrossRefGoogle Scholar
[11]Steinhorn, Ch., A new omitting types theorem, Proceedings of the American Mathematical Society, vol. 89 (1983), pp. 480–486.CrossRefGoogle Scholar