Abstract

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 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 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⊨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⊨Ti} for i = 2, …, 4 is unbounded in κ if κ is indestructibly supercompact. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim