论文标题
Beklemishev的自主性可预测性微积分的可得性和独立性
Deducibility and Independence in Beklemishev's Autonomous Provability Calculus
论文作者
论文摘要
Beklemishev根据可预致性代数的自主扩展引入了Feferman-SchütteOldinal$γ_0$ $γ_0$的顺序符号系统。在本文中，我们介绍逻辑$ \ textbf {bc} $（用于支架色积）。 $ \ textbf {bc} $的语言将所述列表系统扩展到严格的积极模态语言。因此，与其他可预订性逻辑不同，$ \ textbf {bc} $基于一个独立的签名，该签名产生了序数符号系统，而不是先验给出的某些序数所索引的方式。事实证明，所呈现的逻辑等于$ \ textbf {rc} _ {γ_0} $，即$ \ textbf {glp} _ {γ_0} $的严格阳性片段。然后，我们根据$ \ textbf {bc} $来定义一个组合语句，并表明其独立于算术算术递归递归的理论$ \ textbf {atr} _0 $，这是二阶算术理论算术的理论，算术比peano arithmetic强大。
Beklemishev introduced an ordinal notation system for the Feferman-Schütte ordinal $Γ_0$ based on the autonomous expansion of provability algebras. In this paper we present the logic $\textbf{BC}$ (for Bracket Calculus). The language of $\textbf{BC}$ extends said ordinal notation system to a strictly positive modal language. Thus, unlike other provability logics, $\textbf{BC}$ 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 $\textbf{RC}_{Γ_0}$, that is, to the strictly positive fragment of $\textbf{GLP}_{Γ_0}$. We then define a combinatorial statement based on $\textbf{BC}$ and show it to be independent of the theory $\textbf{ATR}_0$ of Arithmetical Transfinite Recursion, a theory of second order arithmetic far more powerful than Peano Arithmetic.
