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Ideals of independence

  • Vera Fischer
  • Diana Carolina MontoyaEmail author
Open Access
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Abstract

We study two ideals which are naturally associated to independent families. The first of them, denoted \(\mathcal {J}_\mathcal {A}\), is characterized by a diagonalization property which allows along a cofinal sequence (the order type of which of uncountable cofinality) of stages along a finite support iteration to adjoin a maximal independent family. The second ideal, denoted \(\mathrm {id}(\mathcal {A})\), originates in Shelah’s proof of \(\mathfrak {i}<\mathfrak {u}\) in Shelah (Arch Math Log 31(6), 433–443, 1992). We show that for every independent family \(\mathcal {A}\), \(\mathrm {id}(\mathcal {A})\subseteq \mathcal {J}_\mathcal {A}\) and define a class of maximal independent families, to which we refer as densely maximal, for which the two ideals coincide. Building upon the techniques of Shelah (1992) we characterize Sacks indestructibility for such families in terms of properties of \(\mathrm {id}(\mathcal {A})\) and devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family.

Keywords

Independent families Sacks indestructibility Constellations of cardinal characteristics 

Mathematics Subject Classification

03E17 03E35 

Notes

Acknowledgements

Open access funding provided by Austrian Science Fund (FWF).

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Kurt Gödel Research CenterUniversity of ViennaViennaAustria

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