2024/12/20 by Levine, Maxwell, Mildenberger, Heike
#03E04 #03E35 #03E55 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2412.16071
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 \squareκ^* holds. In this paper we will present a strongly contrasting result for ℵω. We construct a model in which \squareℵn holds for all n<ω, all scales on ℵω are good, but in which \squareℵω^* fails and some weak forms of internal approachability for [H(ℵω+1)]ℵ1 fail. This requires an extensive analysis of the dominating and approximation properties of a version of Namba forcing. We also prove some supporting results.