The Ultrapower Axiom UA and the number of normal measures over $\aleph_1$ and $\aleph_2$

Abstract

We show that assuming the consistency of certain large cardinals (namely a supercompact cardinal with a measurable cardinal above it of the appropriate Mitchell order) together with the Ultrapower Axiom UA introduced by Goldberg in [3], it is possible to force and construct choiceless universes of 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 $\mu_\omega = \{x \subseteq \aleph_1 \mid x$ contains a club set$\}$, and $\aleph_2$ will carry exactly $\tau$ normal measures, where $\tau = \aleph_n$ for $n=0,1,2$ or $\tau = n$ for $n \ge 1$ an integer (so in particular, $\tau \le \aleph_2$ is any nonzero finite or infinite cardinal). This complements the results of [1] in which $\tau \ge \aleph_3$ and contrasts with the well-known facts that assuming AD + DC, $\aleph_1$ is measurable and carries exactly one normal measure, and $\aleph_2$ is measurable and carries exactly two normal measures.