Analysis of HOD for Admissible Structures

Abstract

Let $n\ge 1$ and assume that there is a Woodin cardinal. For $x\in \mathbb{R}$ let ${\alpha }_x$ be the least $\beta$ such that \[ L_β[x] \models Σ_n \text{-KP} + \exists κ(``κ\text{ is inaccessible and }κ^+ \text{ exists}"). \] 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.

52 pages