## Notre Dame Journal of Formal Logic

### Canjar Filters

#### Abstract

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 Hrušák and Minami. 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 Brendle. 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.

#### Article information

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

Dates
Accepted: 14 October 2013
First available in Project Euclid: 6 April 2016

https://projecteuclid.org/euclid.ndjfl/1459967875

Digital Object Identifier
doi:10.1215/00294527-3496040

Mathematical Reviews number (MathSciNet)
MR3595342

Zentralblatt MATH identifier
06686418

#### Citation

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. https://projecteuclid.org/euclid.ndjfl/1459967875

#### References

• [1] Bartoszyński, T., and H. Judah, Set Theory: On the Structure of the Real Line, A K Peters, Wellesley, Mass., 1995.
• [2] Blass, A., “Combinatorial cardinal characteristics of the continuum,” pp. 395–489 in Handbook of Set Theory, Vols. 1, 2, 3, edited by M. Foreman and A. Kanamori, Springer, Dordrecht, 2010.
• [3] Blass, A., M. Hrušák, and J. Verner, “On strong $P$-points,” Proceedings of the American Mathematical Society, vol. 141 (2013), pp. 2875–83.
• [4] Brendle, J., “Mob families and mad families,” Archive for Mathematical Logic, vol. 37 (1997), pp. 183–97.
• [5] Canjar, R. M., “Mathias forcing which does not add dominating reals,” Proceedings of the American Mathematical Society, vol. 104 (1988), pp. 1239–48.
• [6] Chodounský, D., D. Repovš, and L. Zdomskyy, “Mathias forcing and combinatorial covering properties of filters,” Journal of Symbolic Logic, vol. 80 (2015), pp. 1398–410.
• [7] Guzmán, O., M. Hrušák, and A. Martínez-Celis, “Canjar filters II: Proofs of $\mathfrak{b}<\mathfrak{s}$ and $\mathfrak{b}<\mathfrak{a}$ revisited,” preprint, 2014, http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/195851/1/1895-07.pdf.
• [8] Hrušák, M., and D. Meza-Alcántara, “Comparison game on Borel ideals,” Commentationes Mathematicae Universitatis Carolinae, vol. 52 (2011), pp. 191–204.
• [9] Hrušák, M., and H. Minami, “Mathias-Prikry and Laver-Prikry type forcing,” Annals of Pure and Applied Logic, vol. 165 (2014), pp. 880–94.
• [10] Kechris, A. S., Classical Descriptive Set Theory, vol. 156 of Graduate Texts in Mathematics, Springer, New York, 1995.
• [11] Kechris, A. S., A. Louveau, and W. H. Woodin, “The structure of $\sigma$-ideals of compact sets,” Transactions of the American Mathematical Society, vol. 301 (1987), pp. 263–88.
• [12] Laflamme, C., “Forcing with filters and complete combinatorics,” Annals of Pure and Applied Logic, vol. 42 (1989), pp. 125–63.
• [13] Laflamme, C., and C. C. Leary, “Filter games on $\omega$ and the dual ideal,” Fundamenta Mathematicae, vol. 173 (2002), pp. 159–73.
• [14] Louveau, A., “Sur un article de S. Sirota,” Bulletin des Sciences Mathématiques (2), vol. 96 (1972), pp. 3–7.
• [15] Louveau, A., and B. Velickovic, “Analytic ideals and cofinal types,” Annals of Pure and Applied Logic, vol. 99 (1999), pp. 171–95.
• [16] Mazur, K., “$F_{\sigma}$-ideals and $\omega_{1}\omega_{1}^{*}$-gaps in the Boolean algebras $P(\omega)/I$,” Fundamenta Mathematicae, vol. 138 (1991), pp. 103–11.
• [17] Moore, J. T., M. Hrušák, and M. Džamonja, “Parametrized $\diamondsuit$ principles,” Transactions of the American Mathematical Society, vol. 356 (2004), pp. 2281–306.
• [18] Nyikos, P. J., “Subsets of ${}^{\omega}\omega$ and the Fréchet-Urysohn and $\alpha_{i}$-properties,” Topology and its Applications, vol. 48 (1992), pp. 91–116.
• [19] Sakai, M., and M. Scheepers, “The combinatorics of open covers,” pp. 751–99 in Recent Progress in General Topology, III, edited by K. P. Hart, J. van Mill, and P. Simon, Atlantis Press, Paris, 2014.
• [20] Shelah, S., Proper and Improper Forcing, 2nd ed., vol. 5 of Perspectives in Mathematical Logic, Springer, Berlin, 1998.
• [21] Todorchevich, S., and I. Farah, Some Applications of the Method of Forcing, Yenisei Series in Pure and Applied Mathematics, Yenisei, Moscow, 1995.
• [22] van Mill, J., The Infinite-Dimensional Topology of Function Spaces, vol. 64 of North-Holland Mathematical Library, North-Holland, Amsterdam, 2001.