Some observations on the FGH theorem

TL;DR

This paper explores the FGH theorem in classical and modal propositional logics, analyzing the forms of sentences it guarantees and examining variations related to Rosser provability predicates.

Contribution

It provides new insights into the structure of sentences guaranteed by the FGH theorem and extends the theorem to include variations with Rosser provability predicates.

Findings

Characterization of sentences guaranteed by the FGH theorem

Analysis of FGH theorem forms in classical and modal logics

New variations of the FGH theorem with Rosser provability predicates

Abstract

We investigate the Friedman--Goldfarb--Harrington theorem from two perspectives. Firstly, in the frameworks of classical and modal propositional logics, we study the forms of sentences whose existence is guaranteed by the FGH theorem. Secondly, we prove some variations of the FGH theorem with respect to Rosser provability predicates.