On the number of normal measures $\aleph_1$ and $\aleph_2$ can carry

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.