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MULTIPLE CHOICES IMPLY THE INGLETON AND KREIN–MILMAN AXIOMS

Published online by Cambridge University Press:  12 July 2019

MARIANNE MORILLON
MARIANNE MORILLON*
Affiliation:
LABORATOIRE D’INFORMATIQUE ET MATHÉMATIQUES PARC TECHNOLOGIQUE UNIVERSITAIRE BÂTIMENT 2, 2 RUE JOSEPH WETZELL, 97490 SAINTE CLOTILDE, FRANCE E-mail:marianne.morillon@univ-reunion.fr
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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 (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.

Keywords

Primary 03E25Secondary 46S10Axiom of Choicenonarchimedean fieldsIngleton’s theoremKrein–Milman’s theorem
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 439 - 455
Copyright
Copyright © The Association for Symbolic Logic 2019 

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