MAD Families and SANE Player

2009/04/05 by Saharon Shelah, Shelah, Saharon
Computer Science · Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.0904.0816

openalex publication_date 2009/04/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We throw some light on the question: is there a MAD family (= a family of infinite subsets of N, the intersection of any two is finite) which is completely separable (i.e. any X subseteq N is included in a finite union of members of the family or include a member of the family). We prove that it is hard to prove the consistency of the negation: (a) if 2^aleph₀ < aleph_omega, then there is such a family (b) if there is no such families then some situation related to pcf holds whose consistency is large.

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