Canjar Filters

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 , this solves 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 S f in (Ω,Ω) on subsets of the Cantor space.

[1]  Andreas Blass,et al.  Combinatorial Cardinal Characteristics of the Continuum , 2010 .

[2]  Qi Feng.  Review: S. Todorchevich, I. Farah, Some Applications of the Method of Forcing , 1998 .

[3]  Claude Laflamme,et al.  Filter games on ω and the dual ideal , 2002 .

[4]  Alain Louveau,et al.  Analytic Ideals and Cofinal Types , 1999, Ann. Pure Appl. Log..

[5]  Antje Baer,et al.  The Infinite Dimensional Topology Of Function Spaces , 2016 .

[6]  J. van Mill,et al.  The Infinite-Dimensional Topology of Function Spaces , 2001 .

[7]  M. Džamonja,et al.  PARAMETRIZED ♦ PRINCIPLES , 2003 .

[8]  Krzysztof Mazur,et al.  $F_σ$-ideals and $ω_1 ω^*_1$-gaps in the Boolean algebras Ρ(ω)/I. , 1991 .

[9]  Marion Scheepers,et al.  The Combinatorics of Open Covers , 1996 .

[10]  Lev Bukovský,et al.  The Structure of the Real Line , 2011 .

[11]  Claude Laflamme,et al.  Forcing with Filters and Complete Combinatorics , 1989, Ann. Pure Appl. Log..

[12]  David Meza-Alcántara,et al.  Comparison game on Borel ideals , 2011 .

[13]  Alexander S. Kechris,et al.  The Structure of σ-Ideals of Compact Sets , 1987 .

[14]  P. Nyikos,et al.  Subsets of ωω and the Fréchet-Urysohn and αi-properties , 1992 .

[15]  Lyubomyr Zdomskyy,et al.  MATHIAS FORCING AND COMBINATORIAL COVERING PROPERTIES OF FILTERS , 2014, The Journal of Symbolic Logic.

[16]  Jörg Brendle,et al.  Mob families and mad families , 1998, Arch. Math. Log..

[17]  Benjamin Naumann,et al.  Classical Descriptive Set Theory , 2016 .

[18]  R. Michael Canjar,et al.  Mathias forcing which does not add dominating reals , 1988 .

[19]  Andreas Blass,et al.  On strong P-points , 2013 .