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

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

2009-04-20

arxiv.org/abs/0904.2957v3

Mathematics

Logic

4 pages, no figure

Scientific paper

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.

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