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MAD Saturated Families and SANE Player

Published online by Cambridge University Press:  20 November 2018

Saharon Shelah*
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel, and Department of Mathematics, Hill Center - Busch Campus, Rutgers, The State University of New Jersey, Pis- cataway, NJ 08854-8019, U.S.A. email:
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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 i.e., any $X\,\subseteq \,\mathbb{N}$ 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 ${{2}^{{{\aleph }_{0}}}}\,<\,{{\aleph }_{\omega }}$, 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 ${{a}_{*}}\,>\,{{\aleph }_{1}}$ even unknown);

(iii) if, e.g., there is no inner model with measurables, then there is such a family.

Research Article
Copyright © Canadian Mathematical Society 2011


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