Definability and almost disjoint families

TL;DR

This paper proves the non-existence of infinite maximal almost disjoint families in certain models of set theory, providing new proofs and extending results about definability and maximality of such families.

Contribution

It establishes the non-existence of infinite mad families in Solovay's model and under Martin's Axiom, and offers a new proof regarding analytic mad families.

Findings

No infinite mad families in Solovay's model.

Analytic mad families cannot be maximal.

No $oldsymbol{ ext{Sigma}}^1_2[a]$ mad families if $oldsymbol{ ext{aleph}}_1^{L[a]}<oldsymbol{ ext{aleph}}_1$.

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 $\Sigma^1_2[a]$ infinite mad families.