MAD saturated families and SANE Player

Abstract. 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 U 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 mathamatical equation repersented 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_* > N₁ even unknown);(iii) if, e.g., there is no inner model with measurables, then there is such a family.