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.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 $\kappa <2^{{\mathrm{\aleph }}_0}$, then no $\kappa$-Souslin infinite almost disjoint family can be maximal. Finally we show that if ${\mathrm{\aleph }}_1^{L[a]}<{\mathrm{\aleph }}_1$, then there are no ${\mathrm{\Sigma }}_2^1[a]$ infinite mad families.