On compactness of weak square at singulars of uncountable cofinality

TL;DR

The paper studies compactness of square principles at singular cardinals with uncountable cofinality, aiming to generalize the non-compactness phenomena observed at $\aleph_\omega$. Under mild hypotheses, including a stationary set $S$ with $\square^*_{\delta}$ and a good scale on $\prod_{\delta\in S}\delta^+$, the weak square $\square^*_{\kappa}$ is shown to be compact via a construction of a totally continuous good scale and a fully coherent square sequence. It also shows that strengthening these hypotheses does not force $\square^*_{\aleph_\omega}$, indicating a genuine boundary between the uncountable-cofinality and countable-cofinality cases. These results connect square principles with scales and Aronszajn-tree phenomena, and relate to Golshani's question about Silver-type theorems for special Aronszajn trees.

Abstract

Cummings, Foreman, and Magidor proved that Jensen's square principle is non-compact at $\aleph_ω$, meaning that it is consistent that $\square_{\aleph_n}$ holds for all $n<ω$ while $\square_{\aleph_ω}$ fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild hypotheses, the weak square principle $\square_κ^*$ is in fact compact at singulars of uncountable cofinality, and that an even stronger version of these hypotheses is not enough for compactness of weak square at $\aleph_ω$.

Theorems & Definitions (41)

• theorem 1.1
• proposition 1.2
• proof
• theorem 1.4
• definition 1.5
• definition 1.6
• definition 1.7
• proposition 2.1
• proof
• proposition 2.2: See Remark 11.1 in Cummings2005