We show that base matrices for P(ω)∕fin of regular height larger than h necessarily have maximal branches that are not cofinal. The same holds for base matrices of height h if tSpoiler

Logic and Philosophy of Logic

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The notion of strong measure zero is studied in the context of Polish groups and general separable metric spaces. An extension of a theorem of Galvin, Mycielski and Solovay is given, whereas the theorem is shown to fail for the Baer–Specker group Zω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}^{\omega}}}$$\end{document}. The uniformity number of the ideal of strong measure zero subsets of a separable metric space is examined, providing solutions to several problems of Miller and Steprāns :52–59, 2006).

We show that Hechler’s forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are $${\mathcal {B}}$$ B -Canjar for any countably directed unbounded family $${\mathcal {B}}$$ B of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that $${\mathfrak {b}}=\omega _1$$ b = ω 1 in every extension by the (...) above forcing notions. (shrink)

We show that the model for the simultaneous consistency of the Borel Conjecture and the dual Borel Conjecture given in actually satisfies a stronger version of the dual Borel Conjecture: there are no uncountable very meager sets.