Indestructibility, HOD, and the Ground Axiom
Let ?1 stand for the statement V = HOD and ?2 stand for the Ground Axiom. Suppose Ti for i = 1, , 4 are the theories ZFC + ?1 + ?2, ZFC + ?1 + ?2, ZFC + ?1 + ?2, and ZFC + ?1 + ?2 respectively. We show that if ? is indestructibly supercompact and ? > ? is inaccessible, then for i = 1, , 4, Ai = df{?   ? is inaccessible. We show it is also the case that if ? is indestructibly supercompact, then V??T1, so by reflection, B1 = df{?   ? is inaccessible, we demonstrate that it is possible to construct a model in which ? is indestructibly supercompact and for every inaccessible cardinal ? < ?, V??T1. It is thus not possible to prove in ZFC that Bi = df{? < ??? is an inaccessible limit of inaccessible cardinals and V??Ti} for i = 2, , 4 is unbounded in ? if ? is indestructibly supercompact.  2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
indestructibility hod the ground axiom supercompact cardinal strong cardinal msc (2010) 03e35 03e55