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

T. Mei

Published 2009-04-20, updated 2010-09-08Version 3

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.