HOD in inner models with Woodin cardinals

2020/04/20 by Sandra Müller, Müller, Sandra, Grigor Sargsyan +1
Computer Science · Mathematics · Medicine · #03E45 #03E55 #03E60 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Neurological and metabolic disorders

paper · pdf · doi:10.48550/arxiv.2004.09201

openalex publication_date 2020/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We analyze the hereditarily ordinal definable sets 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 \boldsymbolΠ¹_n+2-determinacy, for a Turing cone of reals x, HOD^M_n(x)[g] = M_n(M_∞ | κ_∞, Λ), where M_∞ is a direct limit of iterates of M_n+1, δ_∞ is the least Woodin cardinal in M_∞, κ_∞ is the least inaccessible cardinal in M_∞ above δ_∞, and Λ is a partial iteration strategy for M_∞. It will also be shown that under the same hypothesis HOD^M_n(x)[g] satisfies GCH.

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