Résume | There is an inherent tension between stationary reflection and the
failure of the singular cardinal hypothesis (SCH). The former is a
compactness type principle that follows from large cardinals.
Compactness is the phenomenon where if a certain property holds for
every smaller substructure of an object, then it holds for the entire
object. In contrast, failure of SCH is an instance of incompactness.
Two classical results of Magidor are:
(1) from large cardinals it is consistent to have reflection at
$\aleph_{\omega+1}$, and
(2) from large cardinals it is consistent to have the failure of SCH at
$\aleph_\omega$.
As these principles are at odds with each other, the natural question
is whether we can have both. We show the answer is yes.
We describe a Prikry style iteration, and use it to force stationary
reflection in the presence of not SCH. Then we obtain this situation at
$\aleph_\omega$ by interleaving collapses. This is joint work with
Alejandro Poveda and Assaf Rinot. |