Powerful types in small theories (Q757349)

The aim of the paper is to suggest an approach to the long-standing problem of existence of a stable theory with more than one but finitely many countable models. Such a theory must be small and, as M. Benda noted, has a non-isolated powerful type. (A theory is small if it has at most countably many types over \(\emptyset\); a type over \(\emptyset\) is called powerful if every model realizing the type realizes every type over \(\emptyset.)\) A. Pillay showed that a small stable theory having a non-isolated powerful type cannot be superstable or 1-based. The author conjectures that such a theory exists; he introduces some (rather technical) notions to express properties of the theory and the type. These properties are formulated in terms of some numerical characteristics which are defined for any type \(p(\bar x)\) and any formula \(\phi(\bar x,\bar y)\) with \(l(\bar x)=l(\bar y)\). A. Pillay proved that, for small 1-based theories, the relation of semi-isolation on the set of realizations of any type is symmetric. (He says that \(\bar a\) semi-isolates \(\bar b\) if there is a formula \(\phi(\bar a,\bar x)\), satisfied by \(\bar b,\) which also determines the pure type of \(\bar b.\)) The author gives an example which shows that one cannot generalize this replacing `1-based' with `stable'. A characterization of powerful types in small theories is also given.