Definability and almost disjoint families

2015/03/25 by Tornquist, Asger · 4 citations
#03E05 #03E15 #03E35 #03E45 #03E50 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1503.07577

Abstract

We show that there are no infinite maximal almost disjoint ("mad") families in Solovay's model, thus solving a long-standing problem posed by A.D.R. Mathias in 1967. We also give a new proof of Mathias' theorem that no analytic infinite almost disjoint family can be maximal, and show more generally that if Martin's Axiom holds at κ<2^ℵ₀, then no κ-Souslin infinite almost disjoint family can be maximal. Finally we show that if ℵ₁^L[a]

Cited by

• Perfect set dichotomy theorem in generalized Solovay model
• The happy coexistence of mad families and Laver measurability
• Combinatorics in Higher Solovay Models
• Independence relations in the Solovay model I

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