Cummings, Foreman, and Magidor investigated the extent to which square principles are compact at singular cardinals. The first author proved that if $\kappa$ is a singular strong limit of uncountable cofinality, all scales on $\kappa$ are good, and $\square^*_\delta$ holds for all $\delta<\kappa$, then $\square_\kappa^*$ holds. In this paper we will present a strongly contrasting result for $\aleph_\omega$. We construct a model in which $\square_{ \aleph_n}$ holds for all $n<\omega$, all scales on $\aleph_\omega$ are good, but in which $\square_{ \aleph_\omega}^*$ fails and some weak forms of internal approachability for $[H(\aleph_{ \omega+1})]^{ \aleph_1}$ fail. This requires an extensive analysis of the dominating and approximation properties of a version of Namba forcing. We also prove some supporting results.
Categories: math.LO
