2011 seminar talk: aleph_1 perfect mad families

Talk held by Jörg Brendle (Kobe University, Japan) at the KGRC seminar on 2011-09-19.

Abstract

We investigate the complexity of maximal almost disjoint (mad) families of subsets of omega. A classical theorem of Mathias says that there are no analytic mad families. On the other hand, Miller proved that there are coanalytic mad families in the constructible universe \(L\). By forcing with a p.o. preserving such a family over \(L\), one sees that the existence of coanalytic mad families is consistent with non-CH. Friedman and Zdomskyy proved that the existence of a \(\Pi^1_2\) mad family is consistent with \(b > \aleph_1\), and asked whether the complexity could be improved to \(\Sigma^1_2\) in their result. In joint work with Yurii Khomskii, we prove that this is indeed the case. (We even conjecture that coanalytic mad families are consistent with \(b > \aleph_1\), though we still do not have a proof for that.) More explicitly, we show that, under CH, one can construct a sequence of \(\aleph_1\) many perfect almost disjoint sets whose union is almost disjoint as well and which survives after adding dominating reals. Under \(V = L\), this sequence, as well as the set defined from it, has a \(\Sigma^1_2\) definition.