Archive for Mathematical Logic 53 (7-8):825-833 (2014)

Abstract
We establish that, in ZF, the statementRLT: Given a setIand a non-empty setF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}}$$\end{document}of non-empty elementary closed subsets of 2Isatisfying the fip, ifF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{F}}$$\end{document}has a choice function, then⋂F≠∅\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bigcap\mathcal{F} \ne \emptyset}$$\end{document},which was introduced in Morillon :739–749, 2012), is equivalent to the Boolean Prime Ideal Theorem. The result provides, on one hand, an affirmative answer to Morillon’s corresponding question in Morillon and, on the other hand, a negative answer—in the setting of ZFA —to the question in Morillon of whether RLT is equivalent to Rado’s selection lemma.
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DOI 10.1007/s00153-014-0390-y
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References found in this work BETA

Some Consequences of Rado’s Selection Lemma.Marianne Morillon - 2012 - Archive for Mathematical Logic 51 (7-8):739-749.
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.
Rado's Selection Lemma Does Not Imply the Boolean Prime Ideal Theorem.Paul E. Howard - 1984 - Mathematical Logic Quarterly 30 (9‐11):129-132.

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Variations of Rado's Lemma.Paul Howard - 1993 - Mathematical Logic Quarterly 39 (1):353-356.
Rado's Selection Lemma Does Not Imply the Boolean Prime Ideal Theorem.Paul E. Howard - 1984 - Mathematical Logic Quarterly 30 (9‐11):129-132.
Some Consequences of Rado’s Selection Lemma.Marianne Morillon - 2012 - Archive for Mathematical Logic 51 (7-8):739-749.
On Vector Spaces Over Specific Fields Without Choice.Paul Howard & Eleftherios Tachtsis - 2013 - Mathematical Logic Quarterly 59 (3):128-146.

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