Proof theory for theories of ordinals III: $\Pi_{N}$-reflection

TL;DR

This paper develops a proof-theoretic framework for theories involving $\Pi_N$-reflection using ordinal diagrams, extending previous work on $\Pi_3$-reflection to higher levels.

Contribution

It introduces a proof-theoretic analysis for $\Pi_N$-reflection theories using ordinal diagrams, generalizing prior results for $\Pi_3$-reflection.

Findings

Provides a proof-theoretic analysis for $\Pi_N$-reflection

Extends ordinal diagram techniques to higher reflection levels

Lays groundwork for further ordinal analysis of reflection principles

Abstract

This paper deals with a proof theory for a theory of $\Pi_{N}$-reflecting ordinals using a system of ordinal diagrams. This is a sequel to the previous one(APAL 129)in which a theory for $\Pi_{3}$-reflection is analysed proof-theoretically.