具有限制实现的可证明性和可解释性逻辑
Provability and interpretability logics with restricted realizations
摘要：
理论T的可证明性逻辑是在任何算术实现下都可在T中证明的模态公式的集合。我们稍微修改了这个概念，要求算术实现来自一个指定的集合$Γ$。我们对可解释性逻辑做了类似的修改。这是2012年的一篇论文。我们首先研究了具有限制实现的可证明性逻辑，并表明对于理论T和限制集$Γ$的各种自然候选者，其中$Γ$中的每个句子在T中具有很好理解的（Meta）数学内容，结果是线性框架的逻辑。然而，对于理论原始递归算术（PRA），我们定义了一个片段，通过利用PRA和I $Σ_1$之间的良好研究关系，产生了一个更有趣的可证明性逻辑。然后，我们研究可解释性逻辑，获得IL（PRA）的一些上界，其特征仍然是一个主要的开放问题，在可解释性逻辑。同样，这个上界与线性标架密切相关。该技术也被应用于产生IL（PRA）$\subset$ ILM的非平凡结果。
英文摘要：
The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $Γ$. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set $Γ$, where each sentence in $Γ$ has a well understood (meta)-mathematical content in T, the result is the logic of linear frames. However, for the theory Primitive Recursive Arithmetic (PRA), we define a fragment that gives rise to a more interesting provability logic, by capitalizing on the well-studied relationship between PRA and I$Σ_1$. We then study interpretability logics, obtaining some upper bounds for IL(PRA), whose characterization remains a major open question in interpretability logic. Again this upper bound is closely relatively to linear frames. The technique is also applied to yield the non-trivial result that IL(PRA) $\subset$ ILM.
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