Abstract.In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, bi-interpretability and mutual interpretability, take the form of isomorphism.
We study the usual notions like monomorphism and product in the various categories. We provide some examples to separate notions across categories. In contrast, we show that, in some cases, notions in different categories do coincide. E.g., we can, under such-and-such conditions, infer synonymity of two theories from their being equivalent in the sense of a coarser equivalence relation.
We illustrate that the categories offer an appropriate framework for conceptual analysis of notions. For example, we provide a ‘coordinate free’ explication of the notion of axiom scheme. Also we give a closer analysis of the object-language/ meta-language distinction.
Our basic category can be enriched with a form of 2-structure. We use this 2-structure to characterize a salient subclass of interpretations, the direct interpretations, and we use the 2- structure to characterize induction. Using this last characterization, we prove a theorem that has as a consequence that, if two extensions of Peano Arithmetic in the arithmetical language are synonymous, then they are identical.
Finally, we study preservation of properties over certain morphisms.
Introduction.Interpretations are ubiquitous in mathematics and logic. Some of the greatest achievements of mathematics, like the internal models of non-euclidean geometries are, in essence, interpretations.
Given the importance of interpretations, it would seem that there is some room for a systematic study of interpretations and interpretability as objects in their own right. This paper is an attempt to initiate one such line of enquiry. It is devoted to the study of the category of interpretations, or, more precisely the study of a sequence of categories of interpretations.
Below, I will briefly address three issues: motivation & desiderata, comparison to some earlier work, and comparison to boolean morphisms.
Motivation & Desiderata.The fact that interpretations play an important role in mathematics does not ipso facto mean that we should study them in a systematic way. Perhaps, as a totality, they are too diverse to make systematic study sensible. Perhaps, the only general insights are trite and not very useful.