The Ultrapower Axiom UA and the number of normal measures over χ_{1} and χ_{2}
A. W. Apter
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 χ_{1} and χ_{2} are both measurable
and carry certain fixed numbers of normal measures.
Specifically, in the models constructed, χ_{1} will carry exactly one normal
measure, namely μ_{ω} ={ x ⊆ χ_{1}  x contains a club set }, and χ_{2}
will carry exactly τ normal measures, where τ = χ_{n} for n = 0,1,2 or
τ = n for n ≥ 1 an integer (so in particular, τ ≤ χ_{2} is any nonzero finite or infinite cardinal).
This complements the results of [1] in which τ ≥ χ_{3} and contrasts with the wellknown facts that assuming
AD + DC, χ_{1} is measurable and carries exactly one normal measure, and χ_{2} is measurable and carries exactly two normal measures.
Tbilisi Mathematical Journal, Vol. 14(1) (2021), pp. 4953
