Abstract view
MAD Saturated Families and SANE Player


Published:20110915
Printed: Dec 2011
Saharon Shelah,
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
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.