There are no minimal effectively inseparable theories

TL;DR

This paper demonstrates that within the framework of effectively inseparable theories, there are no minimal theories with respect to interpretability, highlighting the non-existence of the simplest such theories.

Contribution

It introduces the theory version of effectively inseparable theories (tEI) and proves that no minimal tEI or finitely axiomatizable EI theories exist regarding interpretability.

Findings

No minimal tEI theories exist with respect to interpretability.

tEI is equivalent to EI, confirming the non-existence of minimal EI theories.

Finitely axiomatizable EI theories are also not minimal under interpretability.

Abstract

This paper belongs to the research on the limit of the first incompleteness theorem. Effectively inseparable theories (EI) can be viewed as an effective version of essentially undecidable theories (EU), and EI is stronger than EU. We examine the question: are there minimal effectively inseparable theories with respect to interpretability. We propose tEI, the theory version of EI. We first prove that there are no minimal tEI theories with respect to interpretability (i.e., for any tEI theory $T$, we can effectively find a theory which is tEI and strictly weaker than $T$ with respect to interpretability). By a theorem due to Marian B. Pour-EI, we have tEI is equivalent with EI. Thus, there are no minimal EI theories with respect to interpretability. Also we prove that there are no minimal finitely axiomatizable EI theories with respect to interpretability.