HOD IN INNER MODELS WITH WOODIN CARDINALS. (13th September 2021) - MÜLLER, SANDRA; SARGSYAN, GRIGOR
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 Π ^1_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 .