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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 驴vejdar in his paper (Commentationes Mathematicae Universitatis Carolinae 19:789---813, 1978). The partial solution of 驴vejdar'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.

Interpretability degrees of finitely axiomatized sequential theories