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 אּ₁ and אּ₂ are both measurable and carry certain fixed numbers of normal measures. Specifically, in the models constructed, will carry exactly one normal measure, namely m_w = {x Í אּ₁ | x contains a club set}, and אּ₂ will carry exactly t normal measures, where t ³ אּ₃ is any regular cardinal. This contrasts with the well-known facts that assuming AD + DC, אּ₁ is measurable and carries exactly one normal measure, and אּ₂ is measurable and carries exactly two normal measures.

Tbilisi Mathematical Journal, Vol. 1(2008), pp. 9-14