Refutability as Recursive as Provability
Paola Cattabriga
TL;DR
This paper extends Gödel's arithmetization of provability by introducing a complementary refutation predicate $Rf(x,v)$ alongside the classical $Pf(x,v)$ within PA. It establishes that both predicates are primitive recursive and representable in PA, and it develops lemmas (e.g., notboth) that reveal tight interactions between provability and refutability through their existential counterparts (Bew-like predicates). By analyzing the mutual dependencies between $Pf$ and $Rf$, the work shows how indeterminacy in the incompleteness argument can be reconciled, effectively linking provability and refutability through diagonalization-inspired reasoning. The results contribute a refined perspective on completeness, refutation, and the coding of logical notions in formal systems, with potential implications for decision problems and AI-oriented deduction.
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.