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Multiple Choices imply the Ingleton and Krein-Milman axioms

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Abstract

In set theory without the Axiom of Choice, we consider Ingleton's axiom which is the counterpart in ultrametric analysis of the Hahn-Banach axiom. We show that in $ZFA$, 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 (1992). 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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hal-01982829 , version 1 (16-01-2019)

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Marianne Morillon. Multiple Choices imply the Ingleton and Krein-Milman axioms. 2019. ⟨hal-01982829⟩
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