The nonarithmeticity of the predicate logic of primitive recursive realizability

Abstract

The notion of primitive recursive realizability was introduced by S. Salehi as a kind of semantics for the language of basic arithmetic using primitive recursive functions. It is of interest to study the corresponding predicate logic. D. A. Viter proved that the predicate logic of primitive recursive realizability by Salehi is not arithmetical. The technically complex proof combines the methods used by the author of this article in the study of predicate logics of constructive arithmetic theories and the results of M. Ardeshir on the translation of the intuitionistic predicate logic into the basic predicate logic. The purpose of this article is to present another proof of Viter's result by directly transferring the methods used earlier in proving the nonarithmeticity of the predicate logic of recursive realizability.