Division by zero

TL;DR

This paper demonstrates that the set of Diophantine equations provably unsolvable in Robinson's arithmetic Q is decidable, contrasting with the undecidability in stronger theories, through analysis of a specific class of equations.

Contribution

It introduces a novel approach to decide unsolvability of certain Diophantine equations within Q by analyzing a previously unexplored class of equations and axiomatizing Q's universal fragment.

Findings

Decidability of unsolvable Diophantine equations in Q

Analysis of a new class of Diophantine equations

Axiomatization of Q's universal fragment

Abstract

For any sufficiently strong theory of arithmetic, the set of Diophantine equations provably unsolvable in the theory is algorithmically undecidable, as a consequence of the MRDP theorem. In contrast, we show decidability of Diophantine equations provably unsolvable in Robinson's arithmetic Q. The argument hinges on an analysis of a particular class of equations, hitherto unexplored in Diophantine literature. We also axiomatize the universal fragment of Q in the process.