Projective wellorders and mad families with large continuum
type: Item
instance of: scholarly article
title: Projective wellorders and mad families with large continuum
zbMATH Open document ID: 1267.03049
author: Vera Fischer
author: Sy-David Friedman
author: Lyubomyr Zdomskyy
published in: Annals of Pure and Applied Logic
publication date: 2011-09-22T00:00:00Z
review text: In an earlier paper [Ann. Pure Appl. Logic 161, No. 12, 1581--1587 (2010; Zbl 1225.03059)], the second and the third author investigated how consistently low in the projective hierarchy one can go to find a mad subset of \([\omega]^\omega\) or a mad subset of \(\omega^\omega\), where \([\omega]^\omega\) is the set of all infinite subsets of \(\omega\) and \(\omega^\omega\) is the set of all functions from \(\omega\) to \(\omega\). A family \(A\subseteq [\omega]^\omega\) is called mad (maximal almost disjoint) if (1) any two elements in \(A\) have finite intersection (almost disjoint) and (2) \(A\) is maximal (any \(S\) so that \(A\subsetneq S\subseteq [\omega]^\omega\) must contain two elements that are not almost disjoint). Similarly, \(A\subseteq\omega^\omega\) is mad if (1) any two functions in \(A\) have equal values only on a finite subset of \(\omega\) and (2) any subset of \(\omega^\omega\) which is a proper superset of \(A\) must contain two functions that agree on an infinite set. Now let \(A\subseteq [\omega]^\omega\) be a mad family and define \(\mathcal{L}(A)=\{s\in [\omega]^\omega : s\) is not covered by finitely many \(a\in A\}\). Then \(A\) is said to be \(\omega\)-mad if for every countably infinite \(B\subseteq \mathcal{L}(A)\) there is an \(a\in A\) so that \(|a\cap b|=\omega\) for every \(b\in B\). One defines \(A\subseteq \omega^\omega\) to be \(\omega\)-mad in a similar manner. In [loc. cit.] it was proved that each of the following assertions are consistent with ZFC: 1. \(2^{\aleph_0}=\mathfrak{b}=\omega_2\) and there is a \(\Pi_2^1\)-definable \(\omega\)-mad family \(A\subseteq [\omega]^\omega\). 2. \(2^{\aleph_0}=\mathfrak{b}=\omega_2\) and there is a \(\Pi_2^1\)-definable \(\omega\)-mad family \(A\subseteq \omega^\omega\). The above bounding number \(\mathfrak{b}\) is the smallest cardinality of any family of functions \(F\subseteq \omega^\omega\) whose elements are not eventually bounded by any single \(g\in \omega^\omega\), that is, there …
reviewed by: Daniel W. Cunningham
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E20
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E45
zbMATH DE Number: 5949198
zbMATH Keywords: coding
zbMATH Keywords: projective well-ordering
zbMATH Keywords: projective mad families
zbMATH Keywords: large continuum
MaRDI profile type: Publication
full work available at URL: j.apal.2011.04.001
OpenAlex ID: W2022460137
cites work: Combinatorial Cardinal Characteristics of the Continuum
cites work: Mad families, splitting families and large continuum
cites work: Iterated Forcing and Elementary Embeddings
cites work: A very absolute Π21 real singleton
cites work: Cardinal characteristics and projective wellorders
cites work: Projective mad families
cites work: Long projective wellorderings
DOI: 10.1016/J.APAL.2011.04.001
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description: scientific article; zbMATH DE number 5949198