MAD Saturated Families and SANE Player

by Shelah. [Sh:935]
Canadian J Math, 2011
We throw some light on the question: is there a MAD family (= a family of infinite subsets of N, the intersection of any two is finite) which is completely separable (i.e. any X subseteq N is included in a finite union of members of the family or include a member of the family). We prove that it is hard to prove the consistency of the negation: ``{(a)}'' if 2^{aleph_0} < aleph_omega, then there is such a family ``{(b)}'' if there is no such families then some situation related to pcf holds whose consistency is large.


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