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Countable models of nonmultidimensional ℵ0-stable theories

Published online by Cambridge University Press:  12 March 2014

Elisabeth Bouscaren
Affiliation:
Université Paris VII, 75251 Paris, France
Daniel Lascar
Affiliation:
Université Paris VII, 75251 Paris, France

Extract

In this paper T will always be a countable ℵ0-stable theory, and in this introduction a model of T will mean a countable model.

One of the main notions we introduce is that of almost homogeneous model: we say that a model M of T is almost homogeneous if for all ā and finite sequences of elements in M, if the strong type of ā is the same as the strong type of (i.e. for all equivalence relations E, definable over the empty set and with a finite number of equivalence classes, ā and are in the same equivalence class), then there is an automorphism of M taking ā to . Although this is a weaker notion than homogeneity, these models have strong properties, and it can be seen easily that the Scott formula of any almost homogeneous model is in L1. In fact, Pillay [Pi.] has shown that almost homogeneous models are characterized by the set of types they realize.

The motivation of this research is to distinguish two classes of ℵ0-Stable theories:

(1) theories such that all models are almost homogeneous;

(2) theories with 20 nonalmost homogeneous models.

The example of theories with Skolem functions [L. 1] (almost homogeneous is then equivalent to homogeneous) seems to indicate a link between these properties and the notion of multidimensionality, and that nonmultidimensional theories are in the first case.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1983

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References

[B.]Bouscaren, E., Countable models of multidimensional ℵ0-stable theories, this Journal (to appear).Google Scholar
[L.1]Lascar, D., Les modèles dénombrables d'une théorie ayant des fonctions de Skolem, Transactions of the American Mathematical Society, vol. 268 (1981), pp. 345366.Google Scholar
[L.2]Lascar, D., Ordre de Rudin-Keisler et poids dans les théories stables, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik (to appear).Google Scholar
[L.P.]Lascar, D. and Poizat, B., An introduction to forking, this Journal, vol. 44 (1979), pp. 330350.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
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