Deducibility and independence in Beklemishev's autonomous provability calculus

Abstract

Beklemishev introduced an ordinal notation system for the Feferman-Schütte ordinal Γ0 based on the autonomous expansion of provability algebras. In this paper we present the logic BC (for Bracket Calculus). The language of BC extends said ordinal notation system to a strictly positive modal language. Thus, unlike other provability logics, BC is based on a self-contained signature that gives rise to an ordinal notation system instead of modalities indexed by some ordinal given a priori. The presented logic is proven to be equivalent to RCΓ0, that is, to the strictly positive fragment of GLPΓ0. We then define a combinatorial statement based on BC and show it to be independent of the theory ATR0 of Arithmetical Transfinite Recursion, a theory of second order arithmetic far more powerful than Peano Arithmetic.