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Countable models of multidimensional ℵ0-stable theories

Published online by Cambridge University Press:  12 March 2014

Elisabeth Bouscaren
Elisabeth Bouscaren*
Affiliation:
Université Paris VII, 75251 Paris, France
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Extract

The notations and the definitions used here are identical to those in [B.L.]. We also assume that the reader is familiar with the properties of the Rudin-Keisler order on types [L.].

Let T denote an ℵ0-stable countable theory. The main result in [B.L.] is: All countable models of T are almost homogeneous if and only if T satisfies

(*) For all models M of T and all ā ∈ M, if p ∈ S1(ā) is strongly regular multidimensional, then Dim(p; M) ≥ ℵ0.

In this paper we investigate the case of a theory T which does not satisfy condition (*) and, under certain additional assumptions, we construct nonisomorphic and non-almost-homogeneous countable models. The same type of construction has been used before to show that if T is multidimensional, then for all α ≥ 1, T has at least nonisomorphic models of cardinality ℵα [La.], [Sh.]. As a corollary of our main theorem (Theorem 6) and of the previous result in [B.L.], we prove Vaught's Conjecture (and, in fact, Martin's Strong Conjecture) for theories T with αT finite.

Although we are here interested in countable models, we can also note that our construction proves that theories satisfying the assumptions in Theorem 6 have at least nonisomorphic models of cardinality ℵα.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 2 , June 1983 , pp. 377 - 383
Copyright
Copyright © Association for Symbolic Logic 1983

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References

REFERENCES

[B.L.]Bouscaren, E. and Lascar, D., Countable models of non-multidimensional ℵ0-stable theories, this Journal, vol. 48 (1983), pp. 197–205.Google Scholar
[L.]Lascar, D., Ordre de Rudin-Keisler et poids dans les théories stables, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik (to appear).Google Scholar
[La.]Lachlan, A. H., Spectra of ω-stable theories, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, Band 24, 1978.Google Scholar
[Pi.]Pillay, A., Weakly homogeneous models (to appear).Google Scholar
[Sh.]Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar