$\operatorname {HOD}$ IN INNER MODELS WITH WOODIN CARDINALS. (13th September 2021) - MÜLLER, SANDRA; SARGSYAN, GRIGOR
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 Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol Π ^1_{n+2}$ -determinacy, for a Turing cone of reals x, $\operatorname {HOD} ^{M_n(x)[g]} = M_n(\mathcal {M}_{∞ } | κ _{∞ }, Λ ), $ where $\mathcal {M}_{∞ }$ is a direct limit of iterates of $M_{n+1}$, $δ _{∞ }$ is the least Woodin cardinal in $\mathcal {M}_{∞ }$, $κ _{∞ }$ is the least inaccessible cardinal in $\mathcal {M}_{∞ }$ above $δ _{∞ }$, and $Λ $ is a partial iteration strategy for $\mathcal {M}_{∞ }$ . It will also be shown that under the same hypothesis $\operatorname {HOD}^{M_n(x)[g]} $ satisfies $\operatorname {GCH} $ .