A many-sorted variant of Japaridze’s polymodal provability logic

Abstract

We consider a many-sorted variant of Japaridze's polymodal provability logic GLP. In this variant, propositional variables are assigned sorts n < ω, where variables of sort n are arithmetically interpreted as Πn+1-sentences of the arithmetical hierarchy. We prove that the many-sorted variant is arithmetically complete with respect to this interpretation.The Logics GLP, GLP * , and J * . The polymodal provability logic GLP is formulated in the language of the propositional calculus enriched by unary connectives 0 , 1 , 2 , . . ., called