Forcing indestructibility of MAD families
type: Item
instance of: scholarly article
title: Forcing indestructibility of MAD families
zbMATH Open document ID: 1083.03047
author: Jörg Brendle
author: Shunsuke Yatabe
published in: Annals of Pure and Applied Logic
publication date: 2005-02-23T00:00:00Z
review text: 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 for…
reviewed by: Judith Roitman
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E17
Mathematics Subject Classification ID: 03E40
zbMATH DE Number: 2137211
zbMATH Keywords: maximal almost disjoint families
zbMATH Keywords: tall ideals
zbMATH Keywords: cardinal invariants of the continuum
zbMATH Keywords: Cohen forcing
zbMATH Keywords: random forcing
zbMATH Keywords: Hechler forcing
zbMATH Keywords: Sacks forcing
zbMATH Keywords: Miller forcing
zbMATH Keywords: Laver forcing
zbMATH Keywords: iterated Sacks forcing
zbMATH Keywords: \(\mathbb P\)-indestructilibity
MaRDI profile type: Publication
full work available at URL: j.apal.2004.09.002
OpenAlex ID: W2138117894
cites work: Q4844091
cites work: Adjoining dominating functions
cites work: Combinatorial Cardinal Characteristics of the Continuum
cites work: Regularity properties for dominating projective sets
cites work: Mob families and mad families
cites work: Q4781758
cites work: Q4829656
cites work: Crowded and Selective Ultrafilters under the Covering Property Axiom
cites work: On Tree Ideals
cites work: Q4828855
cites work: Ordering MAD families a la Katětov
cites work: Q4354789
cites work: On non-wellfounded iterations of the perfect set forcing
cites work: Set theory. An introduction to independence proofs
cites work: Q3715106
cites work: Cohen-stable families of subsets of integers
cites work: Hechler reals
cites work: A special class of almost disjoint families
cites work: Two cardinal invariants of the continuum \(({\mathfrak d} <{\mathfrak a})\) and FS linearly ordered iterated forcing
cites work: Dominating projective sets in the Baire space
cites work: Q4848580
cites work: Descriptive set theory and definable forcing
cites work: ISOLATING CARDINAL INVARIANTS
cites work: Towards a Problem of E. van Douwen and A. Miller
DOI: 10.1016/J.APAL.2004.09.002
Recommended article: On the non-existence of mad families
Recommended article: GENERIC EXISTENCE OF MAD FAMILIES
Recommended article: Q3605701
Recommended article: Q4828855
Recommended article: Mad families, forcing and the Suslin hypothesis
Recommended article: RESTRICTED MAD FAMILIES
Recommended article: Q3593911
Recommended article: Mad families, splitting families and large continuum
Recommended article: Indestructibility of ideals and MAD families
Recommended article: MAD families and strategically bounding forcings
description: scientific article; zbMATH DE number 2137211