Interpretability Degrees of Finitely Axiomatized Sequential Theories

Abstract

In this paper we show that the degrees of interpretability of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory —like Elementary Arithmetic EA, IΣ1, or the Gödel-Bernays theory of sets and classes GB— have suprema. This partially answers a question posed by Vıteslav Švejdar in his paper [Šve78]. The partial solution of Švejdar’s problem follows from a stronger fact: the convexity of the degree structure of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory in the degree structure of the degrees of all finitely axiomatized sequential theories. In the paper we also study a related question: the comparison of structures for interpretability and derivability. In how far can derivability mimic interpretability? We provide two positive results and one negative result.