Mad Families and Sane Player Sh935
نویسنده
چکیده
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) which 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: (a) if 2 ℵ 0 < ℵω, then there is such a family (b) if there is no such families then some situation related to pcf holds whose consistency is large; and if a > ℵ 1 even unknown (c) if there is no inner model with measurables then there is such a family.
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The notion of very mad family is a strengthening of the notion of mad family of functions. Here we show existence of very mad families in different contexts.
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