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Definability and almost disjoint families

Törnquist, Asger (2018) Definability and almost disjoint families. Advances in Mathematics, 330 . pp. 61-73. ISSN 0001-8708. doi:10.1016/j.aim.2018.03.005.

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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.R.D. Mathias in 1969. 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^(ℵ0), then no κ-Souslin infinite almost disjoint family can be maximal. Finally we show that if ℵ_1^(L[a])<ℵ_1, then there are no Σ^1_2[a] infinite mad families.

Item Type:Article
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Additional Information:© 2018 Elsevier Inc. Received 30 March 2015, Accepted 23 February 2018, Available online 16 March 2018.
Subject Keywords:Descriptive set theory; Definability; Projective sets; Maximal almost disjoint families; Solovay's model
Classification Code:MSC: 03E05; 03E15; 03E35; 03E45; 03E50
Record Number:CaltechAUTHORS:20180327-083321909
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Official Citation:Asger Törnquist, Definability and almost disjoint families, Advances in Mathematics, Volume 330, 2018, Pages 61-73, ISSN 0001-8708, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85444
Deposited By: Tony Diaz
Deposited On:27 Mar 2018 20:32
Last Modified:15 Nov 2021 20:28

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