On Theories Having a Finite Number of Nonisomorphic Countable Models

?0. Introduction. In this paper we shall state some interesting facts concerning non-w-categorical theories which have only finitely many countable models. Although many examples of such theories are known, almost all of them are essentially the same in the following sense: they are obtained from W-categorical theories, called base theories below, by adding axioms for infinitely many constant symbols. Moreover all known base theories have the (strict) order property in the sense of [6], and so they are unstable. For example, Ehrenfeucht's well-known example which has three countable models has the theory of dense linear order as its base theory. Many papers including [4] and [5] are motivated by the conjecture that every non-w-categorical theory with a finite number of countable models has the (strict) order property, but this conjecture still remains open. (Of course there are partial positive solutions. For example, in [4], Pillay showed that if such a theory has few links (see [1]), then it has the strict order property.) In this paper we prove the instability of the base theory To of such a theory T rather that T itself. Our main theorem is a strengthening of the following which is also our result: if a theory To is stable and w-categorical, then To cannot be extended to a theory T which has n countable models (1 < n < w) by adding axioms for constant symbols.