On definability of types and relative stability

摘要

In this paper, we consider the question of definability of types in non-stable theories. In order to do this we introduce a notion of a relatively stable theory: a theory is stable up to ? if any ?-type over a model has few extensions up to complete types. We prove that an n-type over a model of a theory that is stable up to ? is definable if and only if its ?-part is definable.