Journal of Symbolic Logic 86 (3):871-896 (2021)

Authors
Sandra Eleonore Müller
Ludwig Maximilians Universität, München
Abstract
We analyze the hereditarily ordinal definable sets $\operatorname {HOD} $ in $M_n[g]$ for a Turing cone of reals x, where $M_n$ is the canonical inner model with n Woodin cardinals build over x and g is generic over $M_n$ 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, $\operatorname {HOD} ^{M_n[g]} = M_n,$ 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[g]} $ satisfies $\operatorname {GCH} $.
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DOI 10.1017/jsl.2021.61
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References found in this work BETA

Set-Theoretic Geology.Gunter Fuchs, Joel David Hamkins & Jonas Reitz - 2015 - Annals of Pure and Applied Logic 166 (4):464-501.
The Downward Directed Grounds Hypothesis and Very Large Cardinals.Toshimichi Usuba - 2017 - Journal of Mathematical Logic 17 (2):1750009.
Inner Models with Many Woodin Cardinals.J. R. Steel - 1993 - Annals of Pure and Applied Logic 65 (2):185-209.

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