Journal of Symbolic Logic

Mad families constructed from perfect almost disjoint families

Jörg Brendle and Yurii Khomskii

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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, $\mathfrak{a}_B < \mathfrak{a}$).

Article information

J. Symbolic Logic, Volume 78, Issue 4 (2013), 1164-1180.

First available in Project Euclid: 5 January 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E15: Descriptive set theory [See also 28A05, 54H05] 03E17: Cardinal characteristics of the continuum 03E35: Consistency and independence results

Mad families projective hierarchy cardinal invariants


Brendle, Jörg; Khomskii, Yurii. Mad families constructed from perfect almost disjoint families. J. Symbolic Logic 78 (2013), no. 4, 1164--1180. doi:10.2178/jsl.7804070.

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