The limitless First Incompleteness Theorem

TL;DR

This paper explores the boundaries of the first incompleteness theorem, analyzing the structure of theories where it applies and establishing conditions for the existence of minimal theories with certain properties.

Contribution

It provides a detailed analysis of the Turing and interpretation degree structures of RE theories related to the first incompleteness theorem and generalizes known non-minimality results.

Findings

No minimal essentially undecidable theories with respect to interpretation.

Characterizations of conditions for the non-existence of minimal RE theories with specific properties.

Answers to previously posed questions about the structure of theories satisfying G1.

Abstract

This work is motivated by the problem of 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 [2], 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.