https://hal.univ-reunion.fr/hal-01982829Morillon, MarianneMarianneMorillonLIM - Laboratoire d'Informatique et de Mathématiques - UR - Université de La RéunionMultiple Choices imply the Ingleton and Krein-Milman axiomsHAL CCSD2019[MATH.MATH-LO] Mathematics [math]/Logic [math.LO][MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA]MORILLON, Marianne2019-01-16 08:05:492021-11-03 04:25:302019-01-16 08:05:49enPreprints, Working Papers, ...1In 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.