### 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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