Jonathan Schilhan

A geometric condition for Dependent Choice

The method of permutation models is used to prove basic independence results related to the Axiom of Choice over set theory with atoms. While it is already more than 100 years old (introduced in 1922), arguments that show whether DC holds in a permutation model have always been either ad-hoc or relied simply on the filter of groups being countably closed. We have succeeded in crystallising a condition which characterizes precisely when DC holds or not. Under some circumstances, the same characterization still holds for symmetric extensions.