The Ultrapower Axiom UA and the number of normal measures over χ1 and χ2A. W. ApterWe 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 χ1 and χ2 are both measurable and carry certain fixed numbers of normal measures. Specifically, in the models constructed, χ1 will carry exactly one normal measure, namely μω ={ x ⊆ χ1 | x contains a club set }, and χ2 will carry exactly τ normal measures, where τ = χn for n = 0,1,2 or τ = n for n ≥ 1 an integer (so in particular, τ ≤ χ2 is any nonzero finite or infinite cardinal). This complements the results of [1] in which τ ≥ χ3 and contrasts with the well-known facts that assuming AD + DC, χ1 is measurable and carries exactly one normal measure, and χ2 is measurable and carries exactly two normal measures.Tbilisi Mathematical Journal, Vol. 14(1) (2021), pp. 49-53 |