  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 $\kappa<2^{\aleph_0}$, then no $\kappa$-Souslin
infinite almost disjoint family can be maximal. Finally we show that if
$\aleph_1^{L[a]}<\aleph_1$, then there are no $\Sigma^1_2[a]$ infinite mad
families.
