Refutability as Recursive as Provability

Abstract

Godel numbering is an arithmetization of sintax which defines provability by coding a primitive recursive predicate, Pf(x,v). A multiplicity of researches and results all around this well-known recursive predicate are today widespread in many areas of logic and AI. Not equally investigated is the refutability predicate defined by Godel numbering within the same primitive recursive status. Rf(x,v) can be defined as a recursive predicate meaning that x is the Godel number of a refutation in PA of the formula with Godel number v. This article proposes a logical investigation of the interactive links between provability and refutability predicates when defined within the same recursive status. The resulting Lemmas are clarifying and open new perspectives for the incompleteness argument and the codings of its underlying notions.