We throw some light on the question: is there a MAD family (a maximal family of infinite subsets of ℕ, the intersection of any two is finite) that is saturated (=completely separable i.e., any X ⊆ ℕ is included in a finite union of members of the family or includes a member (and even continuum many members) of the family). We prove that it is hard to prove the consistency of the negation:
(i) if 2ℵ0 < ℵω, then there is such a family;
(ii) if there is no such family, then some situation related to pcf holds whose consistency is large (and if a* > ℵ1 even unknown);
(iii) if, e.g., there is no inner model with measurables, then there is such a family.
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