# Iterates of $M_1$

## TL;DR

This paper explores the relationship between certain inner models and determinacy assumptions, specifically showing an equivalence involving the admissible closure of the Martin-Solovay tree and the direct limit of $M_1$.

## Contribution

It establishes a precise connection between the admissible closure of the Martin-Solovay tree and the direct limit of $M_1$ under $oldsymbol{ riangle}^1_{2}$-determinacy.

## Key findings

- Proves $L_{oldsymbol{
ho}_3}[T_2] igcap V_{u_{oldsymbol{
ho}_3}} = M_{1,oldsymbol{
ho}_3} | u_{oldsymbol{
ho}_3}$.
- Shows the structure of $L_{oldsymbol{
ho}_3}[T_2]$ aligns with the direct limit of $M_1$.
- Provides insights into the hierarchy of inner models under determinacy assumptions.

## Abstract

Assume $\boldsymbol{\Delta}^1_{2}$-determinacy. Let $L_{\kappa_3}[T_2]$ be the admissible closure of the Martin-Solovay tree and let $M_{1,\infty}$ be the direct limit of $M_1$ via countable trees. We show that $L_{\kappa_3}[T_2] \cap V_{u_{\omega}} = M_{1,\infty} | u_{\omega}$.