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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:

03E35 Consistency and independence results
03E55 Large cardinals
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