The interpretability logic of all reasonable arithmetical theories

Feferman 可解释性

Feferman Interpretability

本文引入模态逻辑 FIL 以处理 Feferman 可解释性，通过带标签的模态调整底层理论的公理集，证明了多种可解释性原理在弱算术理论中的可靠性。

A new principle in the interpretability logic of all reasonable arithmetical theories

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 contain…

Interpretability in PRA

In this paper from 2009 we study IL(PRA), the interpretability logic of PRA. As PRA is neither an essentially reflexive theory nor finitely axiomatizable, the two known arithmetical completeness results do not apply to PRA: IL(PRA) is not ILM or ILP. IL(PRA) does of course contain all the principles known to be part of IL(All), the interpretability logic of the principles common to all reasonable arithmetical theories. In this paper, we take two arithmetical properties of PRA and see what their consequences in the modal logic IL(PRA) are. These properties are reflected in the so-called Beklemi…

Provability and interpretability logics with restricted realizations

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. Ho…

关于在皮亚诺算术（PA）中解释的弱集合理论

On Weak Set Theories Interpreted in PA

本文在PA的无参数可解释性下分类弱集合理论，确定其一致性强度，将其分为三类并说明后续研究方向。

弱可解释性逻辑上的持续性原则

The persistence principle over weak interpretability logic

本文研究在弱可解释性逻辑中加入持续性原则所得逻辑，证明其弱不动点性质、切消与 Craig 插值、有限标架性质及算术语义下的可靠完全性。