Hod up to AD$_{\mathbb{R}}+\Theta$ is measurable

TL;DR

This paper investigates the structure of minimal models of AD_R+Theta is measurable, focusing on their HOD and extending previous work to understand their measure-theoretic properties.

Contribution

It extends Trang's work by computing the HOD of minimal models where Theta is measurable under AD_R+.

Findings

Computed HOD of minimal models with measurable Theta

Extended understanding of measure-theoretic properties in AD_R+ models

Provided new insights into the structure of these models

Abstract

Suppose $M$ is a transitive class size model of $AD_{\mathbb{R}}+``\Theta$ is regular". $M$ is a minimal model of $AD_{\mathbb{R}}+``\Theta$ is measurable" if (i) $\mathbb{R}, Ord\subseteq M$ (ii) there is $\mu\in M$ such that $M\models ``\mu$ is a normal $\mathbb{R}$-complete measure on $\Theta$" and (iii) for any transitive class size $N\subsetneq M$ such that $\mathbb{R}\subseteq N$, $N\models ``$there is no $\mathbb{R}$-complete measure on $\Theta$". Continuing Trang's work in [8], we compute HOD of a minimal model of $AD_{\mathbb{R}}+``\Theta$ is measurable".