323 research outputs found

    Menger's covering property and groupwise density

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    We establish a surprising connection between Menger's classical covering property and Blass's modern combinatorial notion of groupwise density. This connection implies a short proof of the groupwise density bound on the additivity number for Menger's property.Comment: Small update

    A semifilter approach to selection principles

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    We develop the semifilter approach to the classical Menger and Hurewicz covering properties and show that the small cardinal g is a lower bound of the additivity number of the family of Menger subspaces of the Baire space, and under u< g every subset X of the real line with the property Split(Lambda,Lambda) is Hurewicz.Comment: LaTeX 2e, 15 pages, submitted to Comment. Math. Univ. Carolina

    Covering the Baire space by families which are not finitely dominating

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    It is consistent (relative to ZFC) that the union of max{b,g} many families in the Baire space which are not finitely dominating is not dominating. In particular, it is consistent that for each nonprincipal ultrafilter U, the cofinality of the reduced ultrapower w^w/U is greater than max{b,g}. The model is constructed by oracle chain condition forcing, to which we give a self-contained introduction.Comment: Small update

    Template iterations with non-definable ccc forcing notions

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    We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if κ\kappa is a measurable cardinal and θ<κ<μ<λ\theta<\kappa<\mu<\lambda are uncountable regular cardinals, then there is a ccc poset forcing s=θ<b=μ<a=λ\mathfrak{s}=\theta<\mathfrak{b}=\mu<\mathfrak{a}=\lambda. Another application is to get models with large continuum where the groupwise-density number g\mathfrak{g} assumes an arbitrary regular value.Comment: To appear in the Annals of Pure and Applied Logic, 45 pages, 2 figure
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