A generalization of a representation of the integers modulo $p$, for the purpose of occasionally establishing the unsolvability of diophantine inequalities

Abstract

It is well known that if a diophantine equation turns out not to have a solution over the integers modulo p, for some p, then it does not have a solution over the integers per se. This is because the integers modulo p are a homomorphic image of the integers. However, the integers modulo p are of little use when faced with diophantine inequalities, as the homomorphic image of the less-than-relation is trivial. The purpose of the present paper is to introduce a way of gereralising a particular representation of the integers modulo p. The generalizations, novel to this paper, are in the form of decidable Lindenbaum-algebras, and allow for deciding whether given positive first-order formulas in the language of first-order arithmetic are solvable. Crucially if a system of diophantine inequalities turns out not to be solvable in one of the Lindenbaum-algebras, then it is not solvable over the standard integers.