Good Scales and Non-Compactness of Squares
Maxwell Levine, Heike Mildenberger
TL;DR
The paper investigates the interaction between good scales and square principles at successors of singular cardinals, focusing on $ ext{aleph}_ ext{ω}$. It develops a two-tier forcing framework using a Laver-style Namba forcing to dominate approximation properties and a forcing to add a good scale, assembled in a revised countable-support iteration to produce a model where $ ext{aleph}_ ext{ω}$ is a strong limit, all scales on $ ext{aleph}_ ext{ω}$ are good, yet $ ext{square}_{ ext{aleph}_ ext{ω}}^{*}$ fails while $ ext{square}_{ ext{aleph}_ ext{n}}$ holds for all $n< ext{ω}$; in addition, there are stationarily-many $N riangleleft H( ext{aleph}_{ ext{ω+1}})$ of size $ ext{aleph}_1$ not sup-internally approachable. The main argument hinges on a delicate lifting of elementary embeddings through the forcing iteration and a careful analysis of the dominating/approximation properties of the Namba-type forcing, together with reflection principles and guessing-model techniques. The work demonstrates a refined boundary between compactness phenomena and internal structure at singulars, and it connects to broader themes about forcing axioms and large cardinals. The results thus illuminate the nuanced balance between below-cardinal square principles and the behavior of scales at $ ext{aleph}_ ext{ω}$.
Abstract
Cummings, Foreman, and Magidor investigated the extent to which square principles are compact at singular cardinals. The first author proved that if $κ$ is a singular strong limit of uncountable cofinality, all scales on $κ$ are good, and $\square^*_δ$ holds for all $δ<κ$, then $\square_κ^*$ holds. In this paper we will present a strongly contrasting result for $\aleph_ω$. We construct a model in which $\square_{\aleph_n}$ holds for all $n<ω$, all scales on $\aleph_ω$ are good, but in which $\square_{\aleph_ω}^*$ fails and some weak forms of internal approachability for $[H(\aleph_{ω+1})]^{\aleph_1}$ fail. This requires an extensive analysis of the dominating and approximation properties of a version of Namba forcing. We also prove some supporting results.