Models of transfinite provability logic

Kripke Models of Transfinite Provability Logic

For any ordinal Λ, we can define a polymodal logic GLPΛ, with a modality [ξ] for each ξ < Λ. These represent provability predicates of increasing strength. Although GLPΛ has no non-trivial Kripke frames, Ignatiev showed that indeed one can construct a universal Kripke frame for the variable-free fragment with natural number modalities, denoted GLPω. In this paper we show how to extend these con...

Models of the Polymodal Provability Logic

The Logic of Provability

This chapter is dedicated to the memory of George Boolos. From the start of the subject until his death on 27 May 1996 he was the prime inspirer of the work in the logic of provability.

Constructive Provability Logic 1

We present two variants of an intuitionistic sequent calculus that validates the Löb rule and permits logical reflection over provability. We explore properties of both variants of this logic, which we call constructive provability logic due to its close ties to Gödel-Löb provability logic.

Constructive Provability Logic

We present constructive provability logic, an intuitionstic modal logic that validates the Löb rule of Gödel and Löb’s provability logic by permitting logical reflection over provability. Two distinct variants of this logic, CPL and CPL*, are presented in natural deduction and sequent calculus forms which are then shown to be equivalent. In addition, we discuss the use of constructive provabili...