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Keywords:

  • Supercompact cardinal;
  • strong cardinal;
  • indestructibility;
  • HOD;
  • the Ground Axiom;
  • MSC (2010) 03E35;
  • 03E55

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∣δ is an inaccessible cardinal which is not a limit of inaccessible cardinals and VδTi} 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 VκT1, so by reflection, B1 = df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and VδT1} 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, VδT1. It is thus not possible to prove in ZFC that Bi = df{δ < κ∣δ is an inaccessible limit of inaccessible cardinals and VδTi} for i = 2, …, 4 is unbounded in κ if κ is indestructibly supercompact. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim