## Journal of Symbolic Logic

### Mad families constructed from perfect almost disjoint families

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

#### Article information

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

Dates
First available in Project Euclid: 5 January 2014

https://projecteuclid.org/euclid.jsl/1388953998

Digital Object Identifier
doi:10.2178/jsl.7804070

Mathematical Reviews number (MathSciNet)
MR3156516

Zentralblatt MATH identifier
1375.03057

#### Citation

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. https://projecteuclid.org/euclid.jsl/1388953998