Modal logical aspects of provability predicates and consistency statements

TL;DR

This paper explores the modal logic properties of provability and consistency in arithmetic theories, establishing arithmetical completeness for several modal logics through advanced techniques.

Contribution

It extends Solovay's method and refines Arai's construction to prove arithmetical completeness of new modal logics related to provability.

Findings

Proves arithmetical completeness of $\\mathsf{NP}$, $\mathsf{ND}$, $\mathsf{NP4}$, and $\mathsf{ND4}$.

Provides a detailed correspondence between derivability conditions and modal logics.

Extends existing methods to new classes of provability predicates.

Abstract

This paper studies the modal logical aspects of provability predicates and consistency statements for theories of arithmetic. First, we provide an overview of previous works on the correspondence between various derivability conditions for provability predicates and different modal logics. The main technical contribution of the present paper is to establish the arithmetical completeness of the logics $\mathsf{NP}$, $\mathsf{ND}$, $\mathsf{NP4}$, and $\mathsf{ND4}$ by extending Solovay's method and refining Arai's construction of Rosser provability predicates.