Extension and Interpretability

Abstract

In this paper we study the combined structure of the relations of theory-extension and interpretability between theories for the case of finitely axiomatised theories. We focus on two main questions. The first is the matter of definability of salient notions in terms of the structure. We show, for example, that local tolerance, locally faithful interpretability and the finite model property are definable over the structure. The second question is how to think about `good' properties of theories that are independent of implementation details and of `bad' properties that do depend on implementation details. Our degree structure is suitable to study this contrast, since one of our basic relations, to wit theory-extension, is dependent on implementation details and the other relation, interpretability, is not. Nevertheless, we can define new good properties using bad ones. We introduce a new notion of sameness of theories \emph{i-bisimilarity} that is second-order definable over our structure. We define a notion of *goodness* in terms of this relation. We call this notion *being fine*. We illustrate that some intuitively good properties, like being a complete theory, are not fine.