The higher sharp II: on $M_2^\#$

TL;DR

This paper explores the descriptive set theoretic representation of the mouse $M_2^{ ext{ extsterling}}$, establishing its equivalence with a certain definability theory involving $L$-indiscernibles, advancing understanding of higher level inner models.

Contribution

It provides a descriptive set theoretic characterization of $M_2^{ ext{ extsterling}}$, linking it to $L_{oldsymbol{ riangle}^1_3}[T_3]$ and higher level indiscernibles, extending previous work on $M_1^{ ext{ extsterling}}$.

Findings

Proves the many-one equivalence of $M_2^{ ext{ extsterling}}$ and $L_{oldsymbol{ riangle}^1_3}[T_3]$.

Establishes the connection between $M_2^{ ext{ extsterling}}$ and higher level $L$-indiscernibles.

Advances the descriptive set theoretic understanding of higher mouse models.

Abstract

We establish the descriptive set theoretic representation of the mouse $M_n^{\#}$, which is called $0^{(n+1)\#}$. This part partially deals with the case $n=2$ by proving the many-one equivalence of $M_2^{\#}$ and the theory of $L_{\boldsymbol{\delta}^1_3}[T_3]$ with the higher level analogs of $L$-indiscernibles.