Analysis of HOD for Admissible Structures

Abstract

Let $n \geq 1$ and assume that there is a Woodin cardinal. For $x \in \mathbb{R}$ let $\alpha_x$ be the least $\beta$ such that \[ L_\beta [x] \models \Sigma_n \text{-KP} + \exists \kappa (``\kappa \text{ is inaccessible and }\kappa^+ \text{ exists}"). \] We adapt the analysis of $\text{HOD}^{L[x,G]}$ as a strategy mouse to $L_{\alpha_x}[x,G]$ for a cone of reals $x$. That is, we identify a mouse $\mathcal{M}^{\text{n-ad}}$ and define a class $H \subseteq L_{\alpha_x}[x,G]$ as a natural analogue of $\text{HOD}^{L[x,G]} \subseteq L[x,G]$, and show that $H = M_\infty[\Sigma_0]$, where $M_\infty$ is an iterate of $\mathcal{M}^{\text{n-ad}}$ and $\Sigma_0$ a fragment of its iteration strategy.