The higher sharp IV: the higher levels

TL;DR

This paper explores the descriptive set theoretic representation of the mouse $M_n^{\#}$, specifically for the case when $n>3$, contributing to the understanding of higher sharp IV levels.

Contribution

It provides a new descriptive set theoretic framework for representing $0^{(n+1)\#}$ for $n>3$, extending previous work on lower levels.

Findings

Established the representation for $0^{(n+1)\#}$ with $n>3$

Extended the descriptive set theoretic analysis to higher sharp IV levels

Enhanced understanding of the structure of mice in set theory

Abstract

We establish the descriptive set theoretic representation of the mouse $M_n^{\#}$, which is called $0^{(n+1)\#}$. This part deals with the case $n>3$.