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Mad Families Constructed from Perfect Almost Disjoint Families

Published online by Cambridge University Press:  12 March 2014

Jörg Brendle  and
Yurii Khomskii
Jörg Brendle
Affiliation:
Graduate School of System Informatics, Kobe University, Rokko-Dai 1-1, NADA, Kobe 657-8501, Japan, E-mail: brendle@kurt.scitec.kobe-u.ac.jp
Yurii Khomskii
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Straße 25, 1090 Wien, Austria, E-mail: yurii@deds.nl
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Abstract

We prove the consistency of together with the existence of a -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 ℵ1-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, defined as the least number of Borel a.d. sets whose union is a mad family. Our proof yields the consistency of (and hence, ).

Keywords

03E1503E1703E35Mad familiesprojective hierarchycardinal invariants
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 4 , December 2013 , pp. 1164 - 1180
Copyright
Copyright © Association for Symbolic Logic 2013

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References

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