Good Scales and Non-Compactness of Squares

TL;DR

This paper explores the non-compactness of square principles at the singular cardinal , showing that certain strong combinatorial properties can hold while others fail, through complex forcing constructions.

Contribution

It constructs a model demonstrating the failure of * while maintaining other square and scale properties, contrasting previous results.

Findings

* fails at  in the constructed model

All scales on  are good in the model

 holds for all _n, but some internal approachability fails

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 $\square^*_\delta$ 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.