On the strong equality between supercompactness and strong compactness

Abstract

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V ⊨ V \models ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V [ G ] ⊨ V[{G}] \models ZFC + GCH in which, (a) (preservation) for κ ≤ λ \kappa \le \lambda regular, if V ⊨ V \models {} “ κ \kappa is λ \lambda supercompact”, then V [ G ] ⊨ V[G] \models {} “ κ \kappa is λ \lambda supercompact” and so that, (b) (equivalence) for κ ≤ λ \kappa \le \lambda regular, V [ G ] ⊨ V[{G}] \models {} “ κ \kappa is λ \lambda strongly compact” iff V [ G ] ⊨ V[{G}] \models {} “ κ \kappa is λ \lambda supercompact”, except possibly if κ \kappa is a measurable limit of cardinals which are λ \lambda supercompact.