nLab
Fraenkel model of ZFA

Idea

The Fraenkel model is a model of the set theory ZFA that doesn’t satisfy the axiom of choice. It was the first example of a permutation model? of set theory.

Description

Fraenkel’s description used the language of material set theory, and indeed most set theorists would give the description of the Fraenkel model using this language, but it can be described quite simply from a structural perspective, and then the original version can be recovered by considering pure sets (allowing atoms).

The model is given by the topos of sets with an action of an open subgroup of the group $(\mathbb{Z}/2{)}^{\mathbb{N}}$ for a certain topology on this group. Open subgroups are the finite-index subgroups ${\prod }_{i\in I}{H}_i\times (\mathbb{Z}/2{)}^{(\mathbb{N}-I)}$ for finite $I\subset \mathbb{N}$ and $H_i\le \mathbb{Z}/2$. Arrows in this topos are allowed to be equivariant for an open (possibly proper) subgroup of the groups acting on the domain and codomain.