Definability and almost disjoint families

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 $\kappa <2^{{\aleph }_0}$, 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 ${\Sigma }_2^1[a]$ infinite mad families.

Changes in version 2: (1) The proof of Claim 2 on p. 9 has been fixed. (2) Notation regarding characteristic functions and the sets they define has been explained more clearly (3) Typos corrected throughout the manuscript