The limitless First Incompleteness Theorem

Abstract: This work is motivated from finding the limit of the applicability of the first incompleteness theorem ($\sf G1$). A natural question is: can we find a minimal theory for which $\sf G1$ holds? We examine the Turing degree structure of recursively enumerable (RE) theories for which $\sf G1$ holds and the interpretation degree structure of RE theories weaker than the theory $\mathbf{R}$ with respect to interpretation for which $\sf G1$ holds. We answer all questions that we posed in \cite{Cheng20}, and prove more results about them. It is known that there are no minimal essentially undecidable theories with respect to interpretation. We generalize this result and give some general characterizations which tell us under what conditions there are no minimal RE theories having some property with respect to interpretation.