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1. CJM 2011 (vol 63 pp. 1416)

Shelah, Saharon
MAD Saturated Families and SANE Player
We throw some light on the question: is there a MAD family (a maximal family of infinite subsets of $\mathbb{N}$, the intersection of any two is finite) that is saturated (completely separable \emph{i.e.,} any $X \subseteq \mathbb{N}$ is included in a finite union of members of the family \emph{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 $2^{\aleph_0} \lt \aleph_\omega$, 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 ${\mathfrak a}_* \gt \aleph_1$ even unknown); (iii) if, \emph{e.g.,} there is no inner model with measurables, \emph{then} there is such a family.

Keywords:set theory, MAD families, pcf, the continuum
Categories:03E05, 03E04, 03E17

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