MAD Saturated Families and SANE Player

Shelah, Saharon

Canadian journal of mathematics, Tome 63 (2011) no. 6, pp. 1416-1435

Résumé

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.

Détail

DOI : 10.4153/CJM-2011-057-1

Mots-clés : 03E05, 03E04, 03E17, set theory, MAD families, pcf, the continuum