Abstract :
We consider cartesian categorical (free-variables) theory PR of primitive recursion and arithmetise (godelise) it into the natural numbers set of a classical set theory T: We evaluate the map codes of the coded theory by a general recursive T map and construct a -recursive decision algorithm based on evaluation of primitive recursive map codes. Within theory T strengthend by p. r. internal inconsistency axiom, the predicate decision algorithm turns out to be total, terminating. It decides in a uniform way all diophantine equations and contradicts within the strengthend theory Matiyasevich's negative solution of Hilbert's 10th problem. But by Godel's second incompleteness theorem the strengthend theory is relative consistent to T: This is to show inconsistency of classical set theorie(s).