Definability and almost disjoint families

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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<2ℵ0, then no κ-Souslin infinite almost disjoint family can be maximal. Finally we show that if ℵL[a]1<ℵ1, then there are no Σ12[a] infinite mad families.
TidsskriftAdvances in Mathematics
Sider (fra-til)61-73
StatusUdgivet - 2018

ID: 184033378