On Hilbert's Tenth Problem

Abstract

Using an iterated Horner schema for evaluation of diophantine polynomials, we define a partial µ-recursive "decision" algorithm decis as a "race" for a first nullstelle versus a first (internal) proof of non-nullity for such a polynomial -within a given theory T extending Peano Arithmétique PA. If T is diophantine sound, i. e. if (internal) provability implies truth -for diophantine formulae -, then the T-map decis gives correct results when applied to the codes of polynomial inequalities D(x 1 , . . . , x m ) = 0. The additional hypothesis that T be diophantine complete (in the syntactical sense) would guarantee in addition termination of decis on these formula, i. e. decis would constitute a decision algorithm for diophantine formulae in the sense of Hilbert's 10th problem. From Matiyasevich's impossibility for such a decision it follows, that a consistent theory T extending PA cannot be both diophantine sound and diophantine complete. We infer from this the existence of a diophantine formulae which is undecidable by T. Diophantine correctness is inherited by the diophantine completion T of T, and within this extension decis terminates on all externally given diophantine polynomials, correctly. Matiyasevich's theoremfor the strengthening T of T -then shows that T, and hence T, cannot be diophantine sound. But since the internal consistency formula Con T for T implies -within PA -diophantine soundness of T, we get PA ⊢ ¬Con T , in particular PA must derive its own internal inconsistency formula.