There Are No Minimal Effectively Inseparable Theories

Abstract

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