Representation theorems for recursively enumerable sets and a conjecture related to poonen’s large subring of $\bf \mathbb{Q} $

Abstract

It is remarked that unsolvability results can often be extended to yield novel “representation” theorems for the set of all recursively enumerable sets. In particular, it is shown that an analysis of the proof of the unsolvability of Hilbert’s 10th problem over Poonen’s large subring of $ \mathbb{Q} $ can provide such a theorem. Moreover, applying that theorem to the case of a simple set leads to a conjecture whose truth would imply the unsolvability of Hilbert’s 10th problem over $ \mathbb{Q} $ .