Beklemishev introduced an ordinal notation system for the Feferman-Schütte ordinal ${\mathrm{\Gamma }}_0$ based on the autonomous expansion of provability algebras. In this paper we present the logic $\mathbf{\text{BC}}$ (for Bracket Calculus). The language of $\mathbf{\text{BC}}$ extends said ordinal notation system to a strictly positive modal language. Thus, unlike other provability logics, $\mathbf{\text{BC}}$ 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 ${\mathbf{RC}}_{{\mathbf{\Gamma }}_{\mathbf{0}}}$, that is, to the strictly positive fragment of ${\mathbf{GLP}}_{{\mathbf{\Gamma }}_{\mathbf{0}}}$. We then define a combinatorial statement based on $\mathbf{\text{BC}}$ and show it to be independent of the theory ${\mathbf{ATR}}_0$ of Arithmetical Transfinite Recursion, a theory of second order arithmetic far more powerful than Peano Arithmetic.

Beklemishev 为 Feferman-Schütte 序数引入了一个序数符号系统${\mathrm{\Gamma }}_0$基于可证明代数的自主扩展。在本文中，我们提出了逻辑$\mathbf{\text{公元前}}$（对于括号微积分）。的语言$\mathbf{\text{公元前}}$将所述序数符号系统扩展到严格的正模态语言。因此，与其他可证明性逻辑不同，$\mathbf{\text{公元前}}$是基于一个自包含的签名，它产生一个序数符号系统，而不是由一些先验给定的序数索引的模态。所提出的逻辑被证明等价于${\mathbf{钢筋混凝土}}_{{\mathbf{\Gamma }}_{\mathbf{0}}}$，也就是严格正片断${\mathbf{GLP}}_{{\mathbf{\Gamma }}_{\mathbf{0}}}$. 然后我们定义一个组合语句$\mathbf{\text{公元前}}$并证明它独立于理论${\mathbf{ATR}}_0$算术超限递归，一种比皮亚诺算术强大得多的二阶算术理论。