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.

中文翻译：

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多项选择暗示英格尔顿和克雷因-米尔曼公理

在没有选择公理的集合论中，我们考虑英格尔顿公理，它是哈恩-巴纳赫公理的超度量对应物。我们表明在ZFA,IE，在没有选择公理被削弱以允许“原子”的集合论中，英格尔顿公理并不意味着选择公理（这解决了ZFAvan Rooij 提出的一个问题，[27]）。我们也证明了在ZFA，“多项选择”公理隐含了 Krein-Milman 公理。我们推断，在ZFA，Hahn-Banach、Ingleton 和 Krein-Milman 公理的合取并不意味着选择公理。