Abstract
We show that assuming the consistency of certain large cardinals (namely a supercompact cardinal with a measurable cardinal above it), it is possible to force and construct choiceless universes of $\mathsf{ZF}$ in which the first two uncountable cardinals $\aleph_1$ and $\aleph_2$ are both measurable and carry certain fixed numbers of normal measures. Specifically, in the models constructed, $\aleph_1$ will carry exactly one normal measure, namely $\mu_w = \{x \subseteq \aleph_1 \mid x$ contains a club set$\}$, and $\aleph_2$ will carry exactly $\tau$ normal measures, where $\tau \ge \aleph_3$ is any regular cardinal. This contrasts with the well-known facts that assuming $\mathsf{AD} + \mathsf{AC}$, $\aleph_1$ is measurable and carries exactly one normal measure, and $\aleph_2$ is measurable and carries exactly two normal measures.
Funding Statement
The author's research was partially supported by PSC-CUNY Grants and CUNY Collaborative Incentive Grants. In addition, the author wishes to thank the referees for helpful comments and suggestions which have been incorporated into the current version of the paper.
Citation
Arthur W. Apter. "On the number of normal measures $\aleph_1$ and $\aleph_2$ can carry." Tbilisi Math. J. 1 9 - 14, 2008. https://doi.org/10.32513/tbilisi/1528768821
Information