Uniqueness and characterization of prime models over sets for totally transcendental first-order theories

THEoEm. If T is a complete first-order totally transcendental theory then over every T-structure A there is a prime model unique up to isomorphism over A. Moreover M is a prime model over A iff: (1) every finite sequence from M realizes an isolated type over A, and (2) there is no uncountable indiscernible set over A in M. The existence of prime models was proved by Morley [3] and their uniqueness for countable A by Vaught [9]. Sacks asked (see Chang and Keisler [1, question 25]) whether the prime model is unique. After proving this I heard Ressayre had proved that every two strictly prime models over any T-structure A are isomorphic, by a strikingly simple proof. From this follows THEOREM. If T is totally transcendental, M a strictly prime model over A then every elementary permutation of A can be extended to an automorphism of M. (The existence of Mfollows by [3].) By our results this holds for any prime model. On the other hand Ressayre's result applies to more theories. For more information see [6, ?OA]. A conclusion of our theorem is the uniqueness of the prime differentially closed field over a differential field. See Blum [8] for the total transcendency of the theory of differentially closed fields. We can note that the prime model Mover A is minimal over A iff in M there is no indiscernible set over A (which is infinite). In order to help the reader, ??l and 2 contain known results which are from Morely [3] (except 2.3, 2.4), with a variation of the definition of rank type. If we define T as totally transcendental iff R(x = x) < c, then the restriction " T is countable" is superfluous. The result of this paper was announced in [6, ?OA.5B] (in more general form) and in [7, Theorem 6]. Notation. Let Tbe a fixed first-order countable complete theory in the language L. For simplicity all the sets and models we shall deal with, will be of cardinality < W, for some high enough cardinal K; and let A be a K-saturated model of T. As every model of T of cardinality < W is isomorphic to an elementary submodel of M, we can deal with them only. (See Morley and Vaught [5], or Chang and Keisler [1] for K-saturated models.) So let M, N denote elementary submodels of A (of cardinality < c), A, B, C sets of elements of M (of cardinality < K), a, b, c elements of a, 5, c finite sequences of elements of M. Let I MI be the set of elements of M, Al the cardinality of A, so I M JJ is the cardinality of M. Let p j, 0 denote formulas