Indestructibility, HOD, and the Ground Axiom

Let φ1 stand for the statement V = HOD and φ2 stand for the Ground Axiom. Suppose Ti for i = 1, . . . , 4 are the theories “ZFC + φ1 + φ2”, “ZFC + ¬φ1 + φ2”, “ZFC + φ1 + ¬φ2”, and “ZFC + ¬φ1 + ¬φ2” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, . . . , 4, Ai =df {δ < κ | δ is an inaccessible cardinal which is not a limit of inaccessible cardinals and Vδ 2 Ti} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing four models in which Ai = ∅ for i = 1, . . . , 4. In each of these models, there is an indestructibly supercompact cardinal κ, and no cardinal δ > κ is inaccessible. We show it is also the case that if κ is indestructibly supercompact, then Vκ 2 T1, so by reflection, B1 =df {δ < κ | δ is an inaccessible limit of inaccessible cardinals and Vδ 2 T1} is unbounded in κ. Consequently, it is not possible to construct a model in which κ is indestructibly supercompact and B1 = ∅. On the other hand, assuming κ is supercompact and no cardinal δ > κ is inaccessible, we demonstrate that it is possible to construct a model in which κ is indestructibly supercompact and for every inaccessible cardinal δ < κ, Vδ 2 T1. It is thus not possible to prove in ZFC that Bi =df {δ < κ | δ is an inaccessible limit of inaccessible cardinals and Vδ 2 Ti} for i = 2, . . . , 4 is unbounded in κ if κ is indestructibly supercompact. ∗2010 Mathematics Subject Classifications: 03E35, 03E55. †