The classification of excellent classes
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abstract
In [9] and [12], Shelah defined a certain type of Scott sentence which he called excellent. He proved, among other things, that if a Scott sentence is excellent and categorical in some uncountable power then it is categorical in all uncountable powers: the analog of the Morley categoricity theorem. Proving such an analog is often the starting point in the classification of a family of classes. Before beginning this classification in the case of excellent Scott sentences, let us say a few words about what this paper is and what it is not.It is not the beginning of a classification theory for complete sentences in where is countable. Although excellence arises in the study of the model theory of Scott sentences, it is not a dividing line in a classification of them. In particular, the assumption of nonexcellence does not yield much information. In fact, in [3] there is an example of a nonexcellent Scott sentence, categorical in ℵ1 which is. not fully categorical. It seems to the second author that a classification of sentences analogous to the classification of first order theories is a long way off and may not be accomplishable in ZFC.This is not to say that the study of excellent Scott sentences (or the class of models of such which we will call excellent classes) is unproductive. Besides its extreme usefulness in [12], Mekler and Shelah have shown that excellence plays a decisive role in the study of almost free algebras (see [7]). Moreover, as the class of ω-saturated models of an ω-stable theory is an example of an excellent class, the study of excellent classes is at least as difficult as the study of first order ω-stable theories.
