Finding the limit of incompleteness II

Abstract

This work is motivated from finding the limit of the applicability of Gödel's first incompleteness theorem (G1): can we find a minimal theory in some sense for which G1 holds? The answer of this question depends on our definition of minimality. We first show that the Turing degree structure of r.e. theories for which G1 holds is as complex as the structure of r.e. Turing degrees. Then we examine the interpretation degree structure of r.e. theories weaker than the theory R for which G1 holds, and answer some open questions about this structure in the literature.