A simplification of the completeness proofs for Guaspari and Solovay's ${\rm R}$.

Abstract

Alternative proofs for Guaspari and Solovay's completeness theorems for R are presented. R is an extension of the provability logic L and was developed in order to study the formal properties of the provability predicate of PA occurring in sentences that may contain connectives for witness comparison. The proof of the Kripke model completeness theorems employs tail models, as introduced by Visser, instead of the more usual finite Kripke models, which makes it possible to derive arithmetical completeness from Kripke model completeness by literally embedding Kripke models into PA. Our arithmetical completenes theorem form a solution to the problem of obtaining a completeness result with respect to a variety of orderings