In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL ; provability of a formula in GL coincides with provability of its arithmetical interpretations of it in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS ; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic. Proof theory for GL has been studied intensively up to the present day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay. In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut-elimination and some standard consequences of it: the interpolation and definability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call Gödel sentences, using a purely proof-theoretical method.

中文翻译：

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真算术可证明性逻辑的证明理论

在 1976 年的一篇经典论文中，Solovay 证明了模态逻辑 GL 的算术完备性；GL 中公式的可证明性与其在 Peano Arithmetic 中的算术解释的可证明性一致。在那篇论文中，他还提供了一个公理系统 GLS 并证明了 GLS 的算术完备性；GLS 中公式的可证明性与其在算术标准模型中的算术解释的真实性一致。GL 的证明理论一直被深入研究至今。然而，到目前为止，除了 Solovay 提供的公理系统之外，还没有开发甚至提出 GLS 的证明理论，这听起来可能有些奇怪。在本文中，我们基于序贯微积分方法开发了 GLS 的证明理论。我们为 GLS 提供了一个连续的演算，并证明了消减和它的一些标准结果：插值和可定义定理。作为切割消除的另一个结果，我们还使用纯粹的证明理论方法证明了 GL 和 GLS 在我们称为 Gödel 句子的特殊形式的公式方面的等价性。