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December 2013 Mad families constructed from perfect almost disjoint families
Jörg Brendle, Yurii Khomskii
J. Symbolic Logic 78(4): 1164-1180 (December 2013). DOI: 10.2178/jsl.7804070

Abstract

We prove the consistency of $\mathfrak{b} > \aleph_1$ together with the existence of a $\Pi^1_1$-definable mad family, answering a question posed by Friedman and Zdomskyy in [7, Question 16]. For the proof we construct a mad family in $L$ which is an $\aleph_1$-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Borel almost-disjointness number $\mathfrak{a}_B$, defined as the least number of Borel a.d. sets whose union is a mad family. Our proof yields the consistency of $\mathfrak{a}_B < \mathfrak{b}$ (and hence, $\mathfrak{a}_B < \mathfrak{a}$).

Citation

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Jörg Brendle. Yurii Khomskii. "Mad families constructed from perfect almost disjoint families." J. Symbolic Logic 78 (4) 1164 - 1180, December 2013. https://doi.org/10.2178/jsl.7804070

Information

Published: December 2013
First available in Project Euclid: 5 January 2014

MathSciNet: MR3156516
zbMATH: 1375.03057
Digital Object Identifier: 10.2178/jsl.7804070

Subjects:
Primary: 03E15, 03E17, 03E35

Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 4 • December 2013
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