Why the theory R is special

Abstract

Is it possible to give coordinate-free characterizations of salient theories? Such characterizations would always involve some notion of sameness of theories: we want to describe a theory modulo a notion of sameness, without having to give an axiomatization in a specific language. Such a characterization could, e.g., be a first order formula in the language of partial preorderings that describes uniquely a degree in a particular structure of degrees of interpretability. Our theory would be contained in this degree. There are very few examples currently known along these lines, except some rather trivial ones. In this paper we provide a non-trivial characterization of Tarski-Mostowski-Robinson’s theory R. The characterization is in terms of the double degree structure of RE degrees of local and global interpretability. Consider the RE degrees of global interpretability that are in the minimal RE degree of local interpretability. These are the global degrees of the RE locally finitely satisfiable theories. We show that these degrees have a maximum and that R is in that maximum. In more mundane terms: an RE theory is locally finite iff it is globally interpretable in R.