Definability and almost disjoint families

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.R.D. Mathias in 1969. 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 ℵ1L[a]<ℵ1, then there are no Σ21[a] infinite mad families.