Some consequences of Rado’s selection lemma

Archive for Mathematical Logic 51 (7-8):739-749 (2012)
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Abstract

We prove in set theory without the Axiom of Choice, that Rado’s selection lemma (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{RL}}$$\end{document}) implies the Hahn-Banach axiom. We also prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{RL}}$$\end{document} is equivalent to several consequences of the Tychonov theorem for compact Hausdorff spaces: in particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{RL}}$$\end{document} implies that every filter on a well orderable set is included in a ultrafilter. In set theory with atoms, the “Multiple Choice” axiom implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf{RL}}$$\end{document}.

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Citations of this work

Three-space type Hahn-Banach properties.Marianne Morillon - 2017 - Mathematical Logic Quarterly 63 (5):320-333.

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References found in this work

[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
The Axiom of Choice.Thomas J. Jech - 1973 - Amsterdam, Netherlands: North-Holland.
Definability of measures and ultrafilters.David Pincus & Robert M. Solovay - 1977 - Journal of Symbolic Logic 42 (2):179-190.
Rado's selection lemma does not imply the Boolean prime ideal theorem.Paul E. Howard - 1984 - Mathematical Logic Quarterly 30 (9‐11):129-132.
Rado's selection lemma does not imply the Boolean prime ideal theorem.Paul E. Howard - 1984 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 30 (9-11):129-132.

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