Undecidable proposition in PA and Diophantine equation

T. Mei

Published 2005-02-15, updated 2007-08-03Version 3

Based on the MRDP theorem concerning the Hilbert tenth problem, there is a corresponding Diophantine equation called proof equation for every formula of the First-order Peano Arithmetic (PA). A formula is provable in PA, if and only if the corresponding proof equation has solution. Based on proof equation, some famous sentences, e.g., the Godel sentence, the Rosser sentence and the Henkin sentence, can be expressed by the form of Diophantine equation. It is proved that for every axiom and theorem in the PA, we can construct actually a corresponding Diophantine equation, for which we know that it has no solution, but this fact cannot be proved in the PA. This means that, for every axiom and theorem in the PA, we can construct actually a corresponding undecidable proposition. Finally, generalizing the idea of proof equation to any mathematical (set theoretical, number theoretical, algebraic, geometrical, topological, et al) proposition, a project translating the task seeking a proof of the mathematical proposition into solving a corresponding Diophantine equation is discussed.