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Modèles saturés et modèles engendrés par des indiscernables

Published online by Cambridge University Press:  12 March 2014

Benoît Mariou
Benoît Mariou*
Affiliation:
Équipe de Logique, U. F. R. de Mathématiques, Université Paris 7, 75251 Paris Cedex 05., France, E-mail: [email protected]
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Abstract

In the early eighties, answering a question of A. Macintyre, J. H. Schmerl ([13]) proved that every countable recursively saturated structure, equipped with a function β encoding the finite functions, is the β-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 ([5]) showed that the saturated model of cardinality ℵ1 of an ω-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 λ-saturated structures which are ACI-models.

The main results are the following. First it is enough to restrict ourselves to ℵ1-saturated structures: if T has an ℵ1-saturated ACI-model then, for every infinite λ, T has a λ-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 ω-stable, (b) T has an ℵ1-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 ℵ1-saturated ACI-model then T doesn't have the independence property.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 1 , March 2001 , pp. 325 - 348
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Chang, C. C. and Keisler, H. J., Model theory, third ed., North-Holland, 1990.Google Scholar
[2]Hodges, W., Model theory, Cambridge University Press, 1993.CrossRefGoogle Scholar
[3]Kaye, R., Indiscernibles, Automorphisms of first-order structures, Clarendon Press, 1994, pp. 257–279.CrossRefGoogle Scholar
[4]Lascar, D., Stability in model theory, Longman, Harlow and John Wiley, New York, 1987.Google Scholar
[5]Lascar, D., Autour de la propriété du petit indice, Proceedings of the London Mathematical Society, vol. 62 (1991), pp. 25–53.CrossRefGoogle Scholar
[6]Mariou, B., Expansions of strongly minimal sets, en préparation.Google Scholar
[7]Mariou, B., Modèles engendrés par des indiscernables et modèles saturés, Thèse de doctorat, Université Paris VII, 1999.Google Scholar
[8]Mariou, B., Uncountable theories with saturated models generated by indiscernibles, en préparation.Google Scholar
[9]Pillay, A., An introduction to stability theory, Oxford University Press, 1983.Google Scholar
[10]Poizat, B., Cours de théorie des modèles, Nur Al-Mantiq Wal-Ma'rifah, 1985.Google Scholar
[11]Ressayre, J. P., Introduction aux modèles récursivement saturés, Séminaire général de logique 1983–84, Publications Mathematiques de l'Université Paris VII, pp. 53–72.Google Scholar
[12]Rosenstein, J. G., Linear orderings, Academic Press, 1982.Google Scholar
[13]Schmerl, J. H., Recursively saturated models generated by indiscernibles, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 99–105.CrossRefGoogle Scholar