Countable models of small stable theories (Q1173999)

In this paper we study countable models of complete stable theories of a countable language. In Section 2 we prove an ``omitting fl-types'' theorem, which is then used to describe a general method for constructing countable models for small stable theories. This method is a generalization of that exhibited by \textit{S. Shelah}, \textit{L. Harrington} and \textit{M. Makkai} [Isr. J. Math. 49, 269-280 (1984; Zbl 0584.03021)] for \(\omega\)-stable theories. A major rule was played there by an important property of \(\omega\)-stable theories -- the existence of a prime model over any set. Specifically: for any model \(M\) of an \(\omega\)- stable theory \(T\) and any set \(A\subset M\) there exists a prime model of the theory \(\text{Th}(M_ A)\). Even superstable theories, let alone stable ones, need not have this property. However, the existence of weakly prime models turns out to be sufficient for the construction of a model omitting a set of fl-types. In Section 3 we present some applications of the omitting fl-types theorem for small superstable theories. A key point in the cited paper, where a proof was presented of Vaught's conjecture on the number of countable models for \(\omega\)-stable theories, was the specification of a special class of \(\omega\)-stable theories, namely, those having ENI-DOP. Every theory in this class has \(2^ \omega\) isomorphism types of countable models. In the present paper this class will be enlarged, considering the class of theories having fl-DOP in the class of all countable superstable theories, and as an application of the omitting fl- types theorem it will be shown that every superstable theory \(T\) having fl-DOP has \(2^ \omega\) isomorphism types of countable models. Another interesting result of this section states, essentially, that a superstable theory with less than \(2^ \omega\) isomorphism types of countable models has fl-depth at most 2.