## On the number of normal measures $$\aleph_1$$ and $$\aleph_2$$ can carry.(English)Zbl 1158.03032

Summary: 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 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 $$\tau\geq\aleph_3$$ s any regular cardinal. This contrasts with the well-known facts that assuming $${\mathbf {AD}}+ {\mathbf{DC}}$$, $$\aleph_1$$ is measurable and carries exactly one normal measure, and $$\aleph_2$$ is measurable and carries exactly two normal measures.

### MSC:

 3e+35 Consistency and independence results 3e+55 Large cardinals