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# MAD Saturated Families and SANE Player

Published:2011-09-15
Printed: Dec 2011
• Saharon Shelah,
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
 Format: LaTeX MathJax

## Abstract

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
 MSC Classifications: 03E05 - Other combinatorial set theory 03E04 - Ordered sets and their cofinalities; pcf theory 03E17 - Cardinal characteristics of the continuum

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