一切合理数论的可解释性逻辑中的一个新原则
A new principle in the interpretability logic of all reasonable arithmetical theories
摘要：
数学理论的可解释性逻辑描述了对该理论的解释的结构行为。不同的理论有不同的逻辑。2011年的这篇论文围绕着这样一个问题：什么样的逻辑描述了所有理论中存在的最少算术量的行为;可以说，所有这些理论的交集。我们用${\textbf{IL}}({\rm All})$表示这个目标逻辑。在本文中，我们提出了一个新的原则$\sf R$在${\textbf{IL}}({\rm All})$。我们表明，$\sf R$不遵循逻辑${\textbf{IL}}{\sf P_0W^*}$，包含所有以前已知的原则。这是通过提供${\textbf{IL}}{\sf P_0W^*}$的模态不完备性证明来完成的：表明$\sf R$在语义上而不是在句法上遵循${\textbf{IL}}{\sf P_0W^*}$。除了用初等方法给出不完备性证明外，我们还简述了如何用所谓的广义Veltman语义学来建立不完备性。在这种程度上，一个新的版本，这种广义Veltman语义的定义和研究。此外，对于重要的原则，框架对应计算。模态结果表明，新原理$\sf R$在任何数论中都是正确的。证明利用了数论中关于可定义割的一些基本结果。
英文摘要：
The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper from 2011 revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by ${\textbf{IL}}({\rm All})$. In this paper we present a new principle $\sf R$ in ${\textbf{IL}}({\rm All})$. We show that $\sf R$ does not follow from the logic ${\textbf{IL}}{\sf P_0W^*}$ that contains all previously known principles. This is done by providing a modal incompleteness proof of ${\textbf{IL}}{\sf P_0W^*}$: showing that $\sf R$ follows semantically but not syntactically from ${\textbf{IL}}{\sf P_0W^*}$. Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with so-called Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated. After the modal results it is shown that the new principle $\sf R$ is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories.
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