Multiple choices imply the ingleton and krein–milman axioms

Journal of Symbolic Logic 85 (1):439-455 (2020)
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Abstract

In set theory without the Axiom of Choice, we consider Ingleton’s axiom which is the ultrametric counterpart of the Hahn–Banach axiom. We show that in ZFA, i.e., in the set theory without the Axiom of Choice weakened to allow “atoms,” Ingleton’s axiom does not imply the Axiom of Choice. We also prove that in ZFA, the “multiple choice” axiom implies the Krein–Milman axiom. We deduce that, in ZFA, the conjunction of the Hahn–Banach, Ingleton and Krein–Milman axioms does not imply the Axiom of Choice.

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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 - 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.
The Hahn-Banach Property and the Axiom of Choice.Juliette Dodu & Marianne Morillon - 1999 - Mathematical Logic Quarterly 45 (3):299-314.

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