2019/01/13 by Morillon, Marianne
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1901.04021
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.