$\operatorname {HOD}$ IN INNER MODELS WITH WOODIN CARDINALS. (13th September 2021)

• Record Type: Journal Article
• Authors: MÜLLER, SANDRA; SARGSYAN, GRIGOR
• Is Part Of: Journal of symbolic logic. Volume 86:Number 3(2021)
• Page Start: 871
• Page End: 896
• Publication Date: 2021-09-13
• Subjects: 03E45 -- 03E60 -- 03E55; HOD -- determinacy -- inner model theory -- large cardinal -- Woodin cardinal -- mouse; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2021.61
• Languages: English
• ISSNs: 0022-4812

Abstract

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 $\boldsymbol \Pi ^1_{n+2}$ -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} $ .