There are no minimal essentially undecidable Theories

TL;DR

This paper proves that within the class of recursively enumerable essentially undecidable theories, no minimal theory exists when ordered by interpretability, highlighting a fundamental complexity in their structure.

Contribution

It establishes the non-existence of a minimal essentially undecidable theory under interpretability, a novel result in the theory of formal systems.

Findings

No minimal essentially undecidable theory exists

The result applies to recursively enumerable theories

Implications for the structure of formal theories

Abstract

We show that there is no theory that is minimal with respect to interpretability among recursively enumerable essentially undecidable theories.