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Journal Articles The Journal of Symbolic Logic Year : 2019

MULTIPLE CHOICES IMPLY THE INGLETON AND KREIN–MILMAN AXIOMS

Abstract

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 (this solves in ZFA a question raised by van Rooij, [27]). 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.

Dates and versions

hal-04524296 , version 1 (28-03-2024)

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Cite

Marianne Morillon. MULTIPLE CHOICES IMPLY THE INGLETON AND KREIN–MILMAN AXIOMS. The Journal of Symbolic Logic, 2019, 85 (1), pp.439-455. ⟨10.1017/jsl.2019.48⟩. ⟨hal-04524296⟩
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