  Using an iterated Horner schema for evaluation of diophantine polynomials, we
define a partial $\mu$-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 Arithmetique 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 $D(x_1,...,x_m) \neq 0$. 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 theorem -- for 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
that PA derives \neg Con_T, in particular PA must derive its own internal
inconsistency formula.

General Mathematics