Proof-theoretic dilator and intermediate pointclasses

TL;DR

This paper explores the connection between two major generalizations of ordinal analysis, demonstrating how $ ext{Sigma}^1_2$-proof theory links Girard's dilators and Pohlers' Spector classes.

Contribution

It reveals the systematic entanglement of these two approaches and highlights the critical role of $ ext{Sigma}^1_2$-proof theoretic analysis in connecting them.

Findings

Demonstrates the systematic entanglement of two major ordinal analysis generalizations.

Shows the critical role of $ ext{Sigma}^1_2$-proof theory in linking Girard's and Pohlers' frameworks.

Provides a unified perspective on ordinal analysis generalizations.

Abstract

There are two major generalizations of the standard ordinal analysis: One is Girard's $\Pi^1_2$-proof theory in which dilators are assigned to theories instead of ordinals. The other is Pohlers' generalized ordinal analysis with Spector classes, where ordinals greater than $\omega_1^{\mathsf{CK}}$ are assigned to theories. In this paper, we show that these two are systematically entangled, and $\Sigma^1_2$-proof theoretic analysis has a critical role in connecting these two.