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Isolating cardinal invariants

Jindřich Zapletal

Journal of Mathematical Logic 3 (1):143-162 (2003)

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Strongly unbounded and strongly dominating sets of reals generalized.Michal Dečo - 2015 - Archive for Mathematical Logic 54 (7-8):825-838.details We generalize the notions of strongly dominating and strongly unbounded subset of the Baire space. We compare the corresponding ideals and tree ideals, in particular we present a condition which implies that some of those ideals are distinct. We also introduce DUI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{DU}_\mathcal{I}}$$\end{document}-property, where I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{I}}$$\end{document} is an ideal on cardinal κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa}$$\end{document}, to capture these two (...) Download Export citation Bookmark
Strongly dominating sets of reals.Michal Dečo & Miroslav Repický - 2013 - Archive for Mathematical Logic 52 (7-8):827-846.details We analyze the structure of strongly dominating sets of reals introduced in Goldstern et al. (Proc Am Math Soc 123(5):1573–1581, 1995). We prove that for every κ (...) Download Export citation Bookmark 2 citations
Van Douwen’s diagram for dense sets of rationals.Jörg Brendle - 2006 - Annals of Pure and Applied Logic 143 (1-3):54-69.details We investigate cardinal invariants related to the structure of dense sets of rationals modulo the nowhere dense sets. We prove that , thus dualizing the already known [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. 183 59–80, Theorem 3.6]. We also show the consistency of each of and . Our results answer four questions of Balcar, Hernández and Hrušák [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. (...) 183 59–80, Questions 3.11]. (shrink) Download Export citation Bookmark 3 citations
The cofinality of the infinite symmetric group and groupwise density.Jörg Brendle & Maria Losada - 2003 - Journal of Symbolic Logic 68 (4):1354-1361.details We show that g ≤ c(Sym(ω)) where g is the groupwise density number and c(Sym(ω)) is the cofinality of the infinite symmetric group. This solves (the second half of) a problem addressed by Thomas. Download Export citation Bookmark 2 citations
Forcing indestructibility of MAD families.Jörg Brendle & Shunsuke Yatabe - 2005 - Annals of Pure and Applied Logic 132 (2):271-312.details Let A[ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P-indestructible yet Q-destructible for several pairs of forcing notions . We close with a detailed investigation of iterated Sacks indestructibility. Download Export citation Bookmark 13 citations
Madness in vector spaces.Iian B. Smythe - 2019 - Journal of Symbolic Logic 84 (4):1590-1611.details We consider maximal almost disjoint families of block subspaces of countable vector spaces, focusing on questions of their size and definability. We prove that the minimum infinite cardinality of such a family cannot be decided in ZFC and that the “spectrum” of cardinalities of mad families of subspaces can be made arbitrarily large, in analogy to results for mad families on ω. We apply the author’s local Ramsey theory for vector spaces [32] to give partial results concerning their definability. Download Export citation Bookmark
Splitting families and forcing.Miloš S. Kurilić - 2007 - Annals of Pure and Applied Logic 145 (3):240-251.details According to [M.S. Kurilić, Cohen-stable families of subsets of the integers, J. Symbolic Logic 66 257–270], adding a Cohen real destroys a splitting family on ω if and only if is isomorphic to a splitting family on the set of rationals, , whose elements have nowhere dense boundaries. Consequently, implies the Cohen-indestructibility of . Using the methods developed in [J. Brendle, S. Yatabe, Forcing indestructibility of MAD families, Ann. Pure Appl. Logic 132 271–312] the stability of splitting families in several (...) Download Export citation Bookmark

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