Year: 2024
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Levine, Maxwell, Mildenberger, Heike
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4405715705