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Mad families and ultrafilters. (English) Zbl 1170.03026
This expository paper presents in great details one of Shelah’s proofs (the one which uses a measurable cardinal) of the consistency of \(\mathfrak{u}<\mathfrak{a}\), where \(\mathfrak{u}\) is the ultrafilter number (i.e., the smallest cardinality of a base of an ultrafilter over \(\omega\)) and \(\mathfrak{a}\) is the almost-disjointness number (i.e., the smallest cardinality of a maximal, infinite set of pairwise almost disjoint infinite subsets of \(\omega\)). At the end of the paper, the author raises some questions which are related to the problem whether the consistency of \(\mathfrak{u}<\mathfrak{a}\) can be carried out in ZFC (i.e., without assuming the existence of a measurable cardinal).

MSC:
03E35 Consistency and independence results
03E05 Other combinatorial set theory
03E17 Cardinal characteristics of the continuum
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