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STRONG COMPACTNESS, SQUARE, GCH, AND WOODIN CARDINALS
Published online by Cambridge University Press: 26 August 2022
Abstract
We show the consistency, relative to the appropriate supercompactness or strong compactness assumptions, of the existence of a non-supercompact strongly compact cardinal
$\kappa _0$
(the least measurable cardinal) exhibiting properties which are impossible when
$\kappa _0$
is supercompact. In particular, we construct models in which
$\square _{\kappa ^+}$
holds for every inaccessible cardinal
$\kappa $
except
$\kappa _0$
, GCH fails at every inaccessible cardinal except
$\kappa _0$
, and
$\kappa _0$
is less than the least Woodin cardinal.
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- © The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic