Indestructibility, HOD, and the Ground Axiom

摘要

are the theories “ZFC + ψ1 + ψ2 ,” “ZFC + -ψ1 +ψ2 ,” “ZFC + ψ1 + -ψ2 ,” and “ZFC + -ψ1 + -ψ2 ”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 - Ti} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate byconstructing via forcing four models in which Ai = θ for i = 1, . . . , 4. In each of these models, there is anindestructibly 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 - T1 , so by reflection, B1 =df {δ < κ | δ is an inaccessible limit of inaccessible cardinals and V_δ - T1 } is unbounded in κ. Consequently, it is not possible to construct a model in which κ is indestructibly supercompact and B1 = θ. 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, V_δ - T_1 . It is thus not possible to prove in ZFC that B_i =df {δ < κ | δ is an inaccessible limit of inaccessible cardinals and V_δ - T_i} for i = 2, . . . , 4 is unbounded in κ if κ is indestructibly supercompact.