Forcing indestructibility of MAD families
A maximal almost disjoint (MAD) family \(\mathcal A\) of subsets of \(\omega\) is said to be \(\mathbb P\)-indestructible iff \(\mathcal A\) remains MAD in \(V^{\mathbb P}\). This paper essentially examines two types of questions: 1. For specific \(\mathbb P\) how can we combinatorially characterize \(\mathbb P\)-indestructibility of MAD families? 2. For a specific \(\mathbb P\) can we consistently find MAD families which are \(\mathbb P\) indestructible, and for specific pairs \(\mathbb P, \mathbb Q\), can we consistently find MAD families which are \(\mathbb P\)-indestructible but not \(\mathbb Q\)-indestructible? It also carefully examines indestructibility for iterated Sacks forcing. As is typical with such questions, the characterization theorems are about tall ideals and the preservation of tallness, where \(\mathcal I\) is tall iff \(\{\omega \setminus I: I \in \mathcal I\}\) has no pseudointersection. This is because an almost disjoint family is MAD iff the ideal it generates is tall. Previous work by Hrušák, Kurilić, and García Ferreira [see \textit{M. Hrušák}, ``Mad families and the rationals'', Commentat. Math. Univ. Carol. 42, 345--352 (2001; Zbl 1051.03039); \textit{M. Hrušák} and \textit{S. García Ferreira}, ``Ordering MAD families à la Katětov'', J. Symb. Log. 68, 1337--1353 (2003; Zbl 1055.03027); \textit{M. Kurilić}, ``Cohen-stable families of subsets of integers'', J. Symb. Log. 66, 257--270 (2001; Zbl 0981.03049)] examined such questions using closed subsets of \(2^{\omega}\) (respectively \(\omega^{\omega}\)) and/or Katětov ordering. This paper uses the notion of \(G_{\delta}\)-closure, where if \(A \subset 2^{< \omega}\) (resp. \(\omega^{< \omega}\)) then the \(G_{\delta}\)-closure of \(A\), \(G_A\), is \(\{x \in 2^{\omega}\) (resp. \(\omega^{< \omega}): \exists^{\infty} n \;x| _n \in A\}\). Its results correct some previously published incorrect characterizations in Hrušák's paper [loc.~cit.], and extend previous results by considering more forcings and such issues as iterations. Most of the characterizations are fairly technical, but there are some simple corollaries, for example a MAD family is Laver-indestructible iff it is indestructible for some forcing which adds a dominating real. The technical characterizations are then used to construct various indestructible MAD families under various set-theoretic hypotheses, e.g., if cov\((\mathcal M) = \mathfrak c\) then there is an Sacks-indestructible MAD family; if add\((\mathcal N) = \mathfrak c\) then there is a random-indestructible MAD family which is Miller-destructible. The forcings considered include Cohen, Hechler, Laver, Miller, random, and Sacks forcings.
scientific article; zbMATH DE number 2137211
scientific article