Ph D thesis - abstract

We consider complete theories, in a first order language, with infinite models. A model of such a theory is said to be weakly Ehrenfeucht if it can be expanded to the algebraic closure of an indiscernible sequence, and is said to be strongly Ehrenfeucht if, in addition, it is saturated in a power which is strictly greater than the cardinality of the expanded language. Given the conflicting combinatorial requirements of these two properties, one naturally asks whether strongly Ehrenfeucht models exist. One also may look for a characterization of theories with strongly Ehrenfeucht models.

In chapters 2 and 3, we show, among other things that: if a theory has a strongly Ehrenfeucht model then it has arbitrarily saturated strongly Ehrenfeucht models; an unstable theory having strong Ehrenfeucht models is not multi-ordered and thus does not have the independence property; the stable theories having strongly Ehrenfeucht models are exactly the superstable theories and every saturated model of a superstable theory is strongly Ehrenfeucht. As well as using properties of indiscernible sequences, we also use some Stability Theory, various combinatorial tools and the model theory of orders - in particular dense orders without endpoints.

In chapter 4, we consider a strongly minimal theory (note that each infinite-dimensional model is a strongly Ehrenfeucht) and we study extensions of this theory in which the algebraic and definable closures coincide; such extensions are called multiplicity-1 extensions. We show that the stability spectrum of multiplicity-1 extensions is affected by the combinatorial properties of the theory: if the theory is not unimodular, its multiplicity-1 extensions are not totally transcendental; if the theory strongly eliminates imaginaries, its multiplicity-1 extensions are not stable.

REFERENCES
A. EHRENFEUCHT & A. MOSTOWSKI, Models of axiomatic theories admitting automorphisms. Fundamenta Mathematicae 43 (1956) 50-68
R. KAYE, Indiscernibles, in Automorphisms of first-order structures, pp 257-279. Clarendon Press, Oxford 1994
D. LASCAR, Autour de la propriété du petit indice. Proceedings of the London Mathematical Society 62 (1991) 25-53
J.P. RESSAYRE, Introduction aux modèles récursivement saturés, in Séminaire général de logique 1983-84, pp 53-72. Publications Mathématiques de l'Université Paris VII
J.H. SCHMERL, Recursively saturated models generated by indiscernibles. Notre Dame Journal of Formal Logic 26 (1985) 99-105

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