Undecidable propositions with Diophantine form arisen from every axiom and every theorem of Peano Arithmetic

TL;DR

This paper demonstrates that every axiom and theorem of Peano Arithmetic can be associated with an undecidable Diophantine proposition, linking proof theory with Diophantine equations and undecidability.

Contribution

It introduces the proof equation concept and universal proof equations for PA, connecting G"odel's theorem with Diophantine forms to establish undecidability.

Findings

Every axiom and theorem of PA corresponds to an undecidable Diophantine proposition.

Constructs a method to transform proof-seeking into solving Diophantine equations.

Links proof theory with Diophantine undecidability through the MRDP theorem.

Abstract

Based on the MRDP theorem, we introduce the ideas of the proof equation of a formula and universal proof equation of Peano Arithmetic (PA); and then, combining universal proof equation and G\"odel's Second Incompleteness Theorem, it is proved that, if PA is consistent, then for every axiom and every theorem of PA, we can construct a corresponding undecidable proposition with Diophantine form. Finally, we present an approach that transforms seeking a proof of a mathematical (set theoretical, number theoretical, algebraic, geometrical, topological, etc) proposition into solving a Diophantine equation.