# HOD in inner models with Woodin cardinals

Research paper by **Sandra Müller, Grigor Sargsyan**

Indexed on: **21 Apr '20**Published on: **20 Apr '20**Published in: **arXiv - Mathematics - Logic**

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#### Abstract

We analyze the hereditarily ordinal definable sets $\operatorname{HOD}$ 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 L\'evy collapse up to its bottom inaccessible cardinal. We
prove that assuming $\boldsymbol\Pi^1_{n+2}$-determinacy, for a Turing cone of
reals $x$, $\operatorname{HOD}^{M_n(x)[g]} = M_n(\mathcal{M}_{\infty} |
\kappa_\infty, \Lambda),$ 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 $\operatorname{HOD}^{M_n(x)[g]}$ satisfies
$\operatorname{GCH}$.