  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.
