# A simplified ordinal analysis of first-order reflection

## TL;DR

This paper presents a simplified ordinal analysis of first-order reflection using a new ordinal notation system based on ψ-functions, and demonstrates how provable Σ₁-sentences are bounded via cut-elimination.

## Contribution

It introduces a simplified ordinal notation system for analyzing first-order reflection and applies cut-elimination to bound provable Σ₁-sentences.

## Key findings

- Ordinal notation system based on ψ-functions
- Bounded provable Σ₁-sentences on L_{ω₁^{CK}}
- Simplified approach to ordinal analysis of reflection

## Abstract

In this note we give a simplified ordinal analysis of first-order reflection. An ordinal notation system $OT$ is introduced based on $\psi$-functions. Provable $\Sigma_{1}$-sentences on $L_{\omega_{1}^{CK}}$ are bounded through cut-elimination on operator controlled derivations.