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__Modèles saturés et modèles engendrés par
des indiscernables - abstract__

In the early eighties, answering a question of A. Macintyre, J.H. Schmerl
([1]) proved that every countable recursively saturated structure, equipped
with a function ß encoding the finite functions, is the ß-closure
of an infinite indiscernible sequence. This result implies that every countably
saturated structure, in a countable but not necessarily recursive language,
is an Ehrenfeucht-Mostowski model, by which we mean that the structure
*expands*,
in a countable language, to the Skolem hull of an infinite indiscernible
sequence (in the new language).
More recently, D. Lascar ([2]) showed that the saturated model of cardinality
*aleph*1
of an *omega*-stable theory is also an Ehrenfeucht-Mostowski model.

These results naturally raise the following problem : which (countable)
complete theories have an uncountably saturated Ehrenfeucht-Mostowski model
? We study a generalization of this question. Namely, we call *ACI-model*
a structure which can be expanded, in a countable language *L'*, to
the algebraic closure (in *L'*) of an infinite indiscernible sequence
(in *L'*). And we try to characterize the *K*-saturated
structures which are ACI-models.

The main results are the following. First it is enough to restrict ourselves
to *aleph*1-saturated structures : if *T* has an *aleph*1-saturated
ACI-model then, for every infinite *K*, s has
a *K*-saturated ACI-model. We obtain a complete
answer in the case of stable theories : if *T* is stable then the
three following properties are equivalent : a) *T* is *omega*-stable,
b) *T* has an *aleph*1-saturated ACI-model, c) every saturated
model of *T* is an Ehrenfeucht-Mostowski model. The unstable case
is more complicated, however we show that if *T* has an *aleph*1-saturated
ACI-model then *T* doesn't have the independence property.

REFERENCES

[1] J.H. SCHMERL, Recursively saturated models
generated by indiscernibles, *Notre Dame Journal of Formal Logic,*
26 (1985) 99-105

[2] D. LASCAR, Autour de la propriété
du petit indice, *Proc. London Math. Soc,* 62 (1991) 25-53

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