# Multiple Choices imply the Ingleton and Krein-Milman axioms

## TL;DR

This paper investigates the relationships between the Ingleton, Krein-Milman, and Axiom of Choice within set theory without Choice, showing that certain axioms do not imply others in $ZFA$ with atoms.

## Contribution

It demonstrates that Ingleton's axiom does not imply the Axiom of Choice and that the multiple Choice axiom implies the Krein-Milman axiom in $ZFA$, clarifying their logical independence.

## Key findings

- Ingleton's axiom does not imply the Axiom of Choice in $ZFA$.
- The multiple Choice axiom implies the Krein-Milman axiom in $ZFA$.
- The conjunction of Hahn-Banach, Ingleton, and Krein-Milman axioms does not imply the Axiom of Choice.

## 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.