Geometric condition for Dependent Choice

TL;DR

This paper introduces a geometric condition that characterizes when the Principle of Dependent Choice holds in certain permutation models, including a new 'nowhere dense' model, with extensions to uncountable cardinals.

Contribution

It presents a novel geometric condition for Dependent Choice in permutation models and introduces the 'nowhere dense' model with its extensions.

Findings

The geometric condition characterizes Dependent Choice in permutation models.

The 'nowhere dense' model satisfies this condition nontrivially.

Extensions of the model to uncountable cardinals are studied.

Abstract

We provide a geometric condition which characterises when the Principle of Dependent Choice holds in a Fraenkel--Mostowski--Specker permutation model. This condition is a slight weakening of requiring the filter of groups to be closed under countable intersections. We show that this condition holds nontrivially in a new permutation model we call "the nowhere dense model" and we study its extensions to uncountable cardinals as well.