HOD in inner models with Woodin cardinals

(with G. Sargsyan)

Submitted. PDF. Bibtex.

Müller, S., & Sargsyan, G. (2018). HOD in inner models with Woodin cardinals.

We analyze the hereditarily ordinal definable sets $\operatorname{HOD}$ in the canonical inner model with $n$ Woodin cardinals $M_n(x,g)$ for a Turing cone of reals $x$, where $g$ is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol\Pi^1_{n+2}$-determinacy, for a Turing cone of reals $x$, $\operatorname{HOD}^{M_n(x,g)} = M_n(M_{\infty}, \Lambda),$ where $M_\infty$ is a direct limit of iterates of an initial segment of $M_{n+1}$ and $\Lambda$ is a partial iteration strategy for $M_{\infty}$. This implies that under the same hypothesis $\operatorname{HOD}^{M_n(x,g)}$ is a fine structural model and therefore satisfies $\operatorname{GCH}$. These results generalize to $\operatorname{HOD}^M$ for self-iterable canonical inner models $M$, for example $M_\omega$, the least mouse with $\omega$ Woodin cardinals, or initial segments of the least non-tame mouse $M_{nt}$.