论文标题
所有合理算术理论的可解释性逻辑的新原则
A new principle in the interpretability logic of all reasonable arithmetical theories
论文作者
论文摘要
数学理论的可解释性逻辑描述了该理论的解释的结构行为。不同的理论具有不同的逻辑。 2011年的本文围绕了以下问题，什么逻辑描述了所有理论中最少算术的行为。可以说，所有这些理论的交叉点。我们用$ {\ textbf {il}}（{\ rm all}）$表示此目标逻辑。在本文中，我们在$ {\ textbf {il}}（{\ rm all}）$中介绍了一个新的原理$ \ sf r $。我们表明，$ \ sf r $并未从逻辑$ {\ textbf {iL}}} {\ sf p_0w^*} $中，其中包含所有先前已知的原则。这是通过提供$ {\ textbf {il}} {\ sf p_0w^*} $的模态不完整的证明来完成的：表明$ \ sf r $在语义上以$ {\ textbf {il textbf {iL}} {i al}} {\ sf p_0w^*} $。除了通过基本方法提供不完整的证明外，我们还素描如何使用所谓的广义Veltman语义来确定不完整。在这个程度上，定义和研究了这种广义Veltman语义的新版本。此外，对于重要的原理，计算框架对应关系。在模态结果之后，结果表明，在任何算术理论中，新原理$ \ sf r $确实有效。该证明对算术理论的可定义削减采用了一些基本结果。
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.
