How many normal measures can $\aleph_{\omega + 1}$ carry?
Volume 191 / 2006
Fundamenta Mathematicae 191 (2006), 57-66
MSC: 03E35, 03E55.
DOI: 10.4064/fm191-1-4
Abstract
We show that assuming the consistency of a supercompact cardinal with a measurable cardinal above it, it is possible for ${\aleph_{\omega + 1}}$ to be measurable and to carry exactly $\tau$ normal measures, where $\tau \ge \aleph_{\omega + 2}$ is any regular cardinal. This contrasts with the fact that assuming AD + DC, ${\aleph_{\omega + 1}}$ is measurable and carries exactly three normal measures. Our proof uses the methods of \cite{AM}, along with a folklore technique and a new method due to James Cummings.
