Definability and almost disjoint families

Asger Tornquist

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arXiv · 1503.07577

Definability and almost disjoint families

Asger Tornquist

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

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BibTeX RIS

Asger Tornquist. 2015-03-29. Definability and almost disjoint families. https://arxiv.org/abs/1503.07577

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