Defining New Universes in Many-sorted Logic. *

In this paper we develop definability theory in such a way that we allow to define new elements also, not only new relations on already existing elements. This is in harmony with our everyday mathematical practice, for example we define new entities when we define a geometry over a field. We will see that, in many respects, defining new elements is more harmonious in many-sorted logic than in one-sorted logic. In the first part of the paper we develop definability theory allowing to define new entities in many-sorted logic (this will amount to defining new universes i.e. new sorts), and in the second part of the paper we develop such a definability theory in one-sorted logic (where this will amount to enlarge the universe with newly defined elements). We will prove an analogon of Beth's definability theorem in this extended context, i.e. we will prove the coincidence of implicit and explicit definability, both in the many-sorted and in the one-sorted case.