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 aleph1 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 aleph1-saturated structures : if T has an aleph1-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 aleph1-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 aleph1-saturated ACI-model then T doesn't have the independence property.

[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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