Good Scales and Non-Compactness of Squares

Abstract

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 ${\square }_{\delta }^{\ast }$ holds for all $\delta <\kappa$, then ${\square }_{\kappa }^{\ast }$ 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 ${\square }_{{\aleph }_{\omega }}^{\ast }$ 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.

Tentative update submitted in September 2025