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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:
 3e+35 Consistency and independence results 300000 Other combinatorial set theory 3e+17 Cardinal characteristics of the continuum
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