Marginalia to two theorems of Magidor

In a paper published at the Annals of Mathematics in 1977,
Magidor proved that it is consistent that the first cardinal at which
the generalized continuum hypothesis (GCH) fails be the very first
singular cardinal Aleph_w. This demonstrates the failure of a
compactness property for Aleph_w. In a paper from 1982, Magidor proved
that Aleph_w can satisfy yet another compactness property. It remained
open ever since whether the two results can coexist in the same model.
In this talk, I will report on a joint work with A. Poveda and D.
Sinapova in which we obtain the desired model.