### On the Strong Equality between Supercompactness and Strong Compactness''

by Apter and Shelah. [ApSh:495]

Transactions American Math Soc, 1997

We show that supercompactness and strong compactness can
be equivalent even as properties of pairs of regular cardinals.
Specifically, we show that if 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] models ZFC + GCH in
which, (a) (preservation) for kappa <= lambda regular, if
V models ``kappa is lambda supercompact'', then V[G]
models ``kappa is lambda supercompact'' and so that, (b)
(equivalence) for kappa <= lambda regular, V[G] models
``kappa is lambda strongly compact'' iff V[G] models
``kappa is lambda supercompact'', except possibly if
kappa is a measurable limit of cardinals which are lambda
supercompact.

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