Instances of dependent choice and the measurability of \(\aleph _{\omega +1}\) (Q1896612)

Strating from cardinals \(\kappa < \lambda\) such that \(\kappa\) is \(\lambda\)-supercompact and \(\lambda\) is measurable, the authors construct a model for the theory ``\(\text{ZF} + \text{DC}_{\aleph_\omega} + \aleph_{\omega + 1}\) is a measurable cardinal''. (Recall that \(\text{DC}_\kappa\) asserts that given a set \(X\) and \(R \subseteq \left(\bigcup_{\alpha \in \kappa} {^\alpha\kappa}\right) \times X\) such that for every \(g \in \bigcup_{\alpha \in \kappa} {^\alpha \kappa}\), there is an \(x\in X\) with \((g,x) \in R\), there is an \(f \in {^\kappa X}\) such that for every \(\alpha \in \kappa\), \((f|\alpha,\;f(\alpha)) \in R)\). The authors point out that \(\text{DC}_{\aleph_{\omega + 1}}\) is inconsistent with the measurability of \(\aleph_{\omega + 1}\), and that by a result of Shelah, if \(\kappa\) is a singular cardinal such that \(\kappa^+\) is measurable and DC\(_\kappa\) holds, then \(\text{cof}(\kappa) = \omega\).