Dilator-based analysis of KP

Abstract

Proof theorists developed various frameworks to analyze impredicative systems like ${\Pi }_1^1\text{-}{\mathsf{CA}}_0$ or $\mathsf{KP}$; One is an operator-controlled derivation system, and the other is Girard's dilator-based $\beta$-logic. In this paper, we provide a functorial formulation of operator-controlled analysis of $\mathsf{KP}$, thereby unifying the two approaches into a single framework. As an application, a new proof of Girard's boundedness theorem is also provided, which states that every ${\Sigma }_1$-over-$L_{{\omega }_1^{\mathsf{CK}}}$-definable function ${\omega }_1^{\mathsf{CK}}\to {\omega }_1^{\mathsf{CK}}$ is bounded by a recursive dilator.

33 pages