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Canjar filters. (English) Zbl 1417.03247

Summary: If \(\mathcal{F}\) is a filter on \(\omega\), we say that \(\mathcal{F}\) is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is \(F_\sigma\), solving a problem of M. Hrušák and H. Minami [Ann. Pure Appl. Logic 165, No. 3, 880–894 (2014; Zbl 1306.03023)]. We give several examples of Canjar and non-Canjar filters; in particular, we construct a \(\mathsf{MAD}\) family such that the corresponding Mathias forcing adds a dominating real. This answers a question of J. Brendle [Arch. Math. Logic 37, No. 3, 183–197 (1998; Zbl 0905.03034)]. Then we prove that in all the “classical” models of \(\mathsf{ZFC}\) there are \(\mathsf{MAD}\) families whose Mathias forcing does not add a dominating real. We also study ideals generated by branches, and we uncover a close relation between Canjar ideals and the selection principle \(S_{\mathrm{fin}}(\Omega,\Omega)\) on subsets of the Cantor space.

MSC:

03E05 Other combinatorial set theory
03E17 Cardinal characteristics of the continuum
03E35 Consistency and independence results
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