Projective wellorders and mad families with large continuum
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 is no \(g\) so that for each \(f\in F\) the set \(\{x\in\omega : f(x)\leq g(x)\}\) is cofinite. At the end of [loc. cit.], the authors of that paper asked the question: Is a projective \(\omega\)-mad family consistent with \(\mathfrak{b}\geq \omega_3\)? In the paper under review, the authors answer this question in the affirmative by proving that the following is consistent with ZFC: \(2^{\aleph_0}=\mathfrak{b}=\omega_3\), there is a \(\Delta_3^1\) well-ordering of the reals, and there is a \(\Pi_2^1\)-definable \(\omega\)-mad family \(A\subseteq [\omega]^\omega\) (resp. \(\omega^\omega\)).
scientific article; zbMATH DE number 5949198
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