# Multiple Choices imply the Ingleton and Krein-Milman axioms

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.
Document type :
Preprints, Working Papers, ...
Domain :

http://hal.univ-reunion.fr/hal-01982829
Contributor : Marianne Morillon <>
Submitted on : Wednesday, January 16, 2019 - 8:05:49 AM
Last modification on : Thursday, March 28, 2019 - 11:24:10 AM

### Identifiers

• HAL Id : hal-01982829, version 1
• ARXIV : 1901.04021

### Citation

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

Record views