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Beklemishev introduced an ordinal notation system for the Feferman-Sch\"utteordinal $\Gamma_0$ based on the autonomous expansion of provability algebras.In this paper we present the logic $\textbf{BC}$ (for Bracket Calculus). Thelanguage of $\textbf{BC}$ extends said ordinal notation system to a strictlypositive modal language. Thus, unlike other provability logics, $\textbf{BC}$is based on a self-contained signature that gives rise to an ordinal notationsystem instead of modalities indexed by some ordinal given a priori. Thepresented logic is proven to be equivalent to $\textbf{RC}_{\Gamma_0}$, thatis, to the strictly positive fragment of $\textbf{GLP}_{\Gamma_0}$. We thendefine a combinatorial statement based on $\textbf{BC}$ and show it to beindependent of the theory $\textbf{ATR}_0$ of Arithmetical TransfiniteRecursion, a theory of second order arithmetic far more powerful than PeanoArithmetic.

Deducibility and Independence in Beklemishev's Autonomous Provability Calculus