HOD in inner models with Woodin cardinals

Abstract

We analyze the hereditarily ordinal definable sets $\mathrm{HOD}$ in $M_n(x)[g]$ for a Turing cone of reals $x$, where $M_n(x)$ is the canonical inner model with $n$ Woodin cardinals build over $x$ and $g$ is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming ${\mathbf{\Pi }}_{n+2}^1$-determinacy, for a Turing cone of reals $x$, ${\mathrm{HOD}}^{M_n(x)[g]}=M_n({\mathcal{M}}_{\infty }|{\kappa }_{\infty },\Lambda ),$ where ${\mathcal{M}}_{\infty }$ is a direct limit of iterates of $M_{n+1}$, ${\delta }_{\infty }$ is the least Woodin cardinal in ${\mathcal{M}}_{\infty }$, ${\kappa }_{\infty }$ is the least inaccessible cardinal in ${\mathcal{M}}_{\infty }$ above ${\delta }_{\infty }$, and $\Lambda$ is a partial iteration strategy for ${\mathcal{M}}_{\infty }$. It will also be shown that under the same hypothesis ${\mathrm{HOD}}^{M_n(x)[g]}$ satisfies $\mathrm{GCH}$.

30 pages