Archive for Mathematical Logic 58 (5-6):767-785 (2019)

Abstract
We study two ideals which are naturally associated to independent families. The first of them, denoted \, is characterized by a diagonalization property which allows along a cofinal sequence of stages along a finite support iteration to adjoin a maximal independent family. The second ideal, denoted \\), originates in Shelah’s proof of \ in Shelah, 433–443, 1992). We show that for every independent family \, \\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 we characterize Sacks indestructibility for such families in terms of properties of \\) and devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family.
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DOI 10.1007/s00153-019-00669-8
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References found in this work BETA

Mad Families, Splitting Families and Large Continuum.Jörg Brendle & Vera Fischer - 2011 - Journal of Symbolic Logic 76 (1):198 - 208.
The Spectrum of Independence.Vera Fischer & Saharon Shelah - 2019 - Archive for Mathematical Logic 58 (7-8):877-884.
Con(U>I).Saharon Shelah - 1992 - Archive for Mathematical Logic 31 (6):433-443.

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