A small ultrafilter number at every singular cardinal (Q6061155)

Let \(\kappa\) be an infinite cardinal and let \(\mathcal{U}\) be a uniform ultrafilter over \(\kappa\). Then:\par --a base for \(\mathcal{U}\) is a collection \(\mathcal{B}\subset \mathcal{U}\) such that for every \(U\in \mathcal{U}\) there is \(B\in \mathcal{B}\) such that \(B\subset^\ast U\) namely, \(B\setminus U\) is a bounded subset of \(\kappa\);\par --the characteristic of \(\mathcal{U}\) is Ch\((\mathcal{U}):=\min\{ \vert \mathcal{B}\vert \colon \mathcal{B}\subset\mathcal{U} \; \text{is a base for} \; \mathcal{U}\}\);\par --the ultrafilter number of \(\kappa\) is defined by \[\mathfrak{u}_\kappa:=\min\{ \text{Ch}(\mathcal{U})\colon \mathcal{U} \;\text{ is a uniform ultrafilter over}\; \kappa\}.\] The number \(\mathfrak{u}_\kappa\) is a generalized characteristic cardinal of the continuum. In ZFC the inequalities \(\kappa^+\le \mathfrak{u}_\kappa\le 2^\kappa\) hold for every infinite cardinal \(\kappa\). Other results are obtained in models of set theory using the iterated forcing method. They may be divided into two groups, one assuming that kappa is regular, see e.g. \textit{D. Raghavan} and \textit{S. Shelah} [Arch. Math. Logic 59, No. 3--4, 325--334 (2020; Zbl 1481.03055))], and the other when \(\kappa\) is a singular cardinal, see \textit{S. Garti} et al., [Acta Math. Hung. 160, No. 2, 320--336 (2020; Zbl 07200100)], or \textit{M. Giti} [Acta Math. Hung. 162, No. 1, 325--332 (2020; Zbl 1474.03124)]. The paper under review belongs to the second group. At the first part, the authors construct a model with a a small ultrafilter number of \(\aleph_{\omega_1}\) namely, in which \[\mathfrak{u}_{\aleph_{\omega_1}}=\aleph_{\omega_1+1}<2^{\aleph_{\omega_1}}=\aleph_{\omega_1+2}.\] In the next sections the authors introduce and develop a version of the overlapping strong extender forcing with collapses which can keep the top cardinal \(\kappa\) inaccessible. Next, they apply this forcing to construct a model where \(\kappa\) is the least inaccessible and \(V_\kappa\) is a model of GCH at regulars, failures of the Singular Cardinal Hypothesis SCH at singulars, and the ultrafilter numbers at all singulars are small.