Maxwell Levine:
Compactness for weak square at singulars of uncountable cofinality
Zeit und Ort
Dienstag, 26.7.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Zusammenfassung
We study versions of Jensen's square principle \(\bsquare_\bkappa\), a combinatorial principle that holds for all cardinals \(\bkappa\) in Gödel's constructible universe \(L\). Cummings, Foreman, and Magidor proved that the square principle is non-compact at \(\baleph_\bomega\), meaning that it is consistent that \(\bsquare_{\baleph_n}\) holds for all \(n<\bomega\) while\n\(\bsquare_{\baleph_\bomega}\) fails. We investigate the natural question of whether this phenomenon generalizes for singulars of uncountable cofinality. Surprisingly, we show that under some mild hypotheses, the weak square principle \(\bsquare_\bkappa^*\) 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 \(\baleph_\bomega\).