Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\kappa<2^{\aleph_0}$, 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.

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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ph
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The categories of this paper are: supr-con
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-ex