Notre Dame Journal of Formal Logic

Canjar Filters

Osvaldo Guzmán, Michael Hrušák, and Arturo Martínez-Celis

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If F is a filter on ω, we say that F is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is Fσ, solving a problem of Hrušák and Minami. We give several examples of Canjar and non-Canjar filters; in particular, we construct a MAD family such that the corresponding Mathias forcing adds a dominating real. This answers a question of Brendle. Then we prove that in all the “classical” models of ZFC there are 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 Sfin(Ω,Ω) on subsets of the Cantor space.

Article information

Notre Dame J. Formal Logic, Volume 58, Number 1 (2017), 79-95.

Received: 1 September 2012
Accepted: 14 October 2013
First available in Project Euclid: 6 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Canjar filters Mathias forcing dominating reals MAD families


Guzmán, Osvaldo; Hrušák, Michael; Martínez-Celis, Arturo. Canjar Filters. Notre Dame J. Formal Logic 58 (2017), no. 1, 79--95. doi:10.1215/00294527-3496040.

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