The set theory with atoms ([ZFA](https://ncatlab.org/nlab/show/ZFA)), is a modified version of set theory, and is characterized by the fact that it admits objects other than sets, atoms. Atoms are objects which do not have any elements.


I have thusfar found two applications of ZFA-Set Theory. Firstly, ZFA can be used to [prove](https://www.sciencedirect.com/science/article/pii/S0049237X08717567) the independence of the Axiom of Choice.


Secondly, one can show a [correspondence](https://mathoverflow.net/questions/119667/models-of-zfa-corresponding-exactly-with-a-particular-class-of-groups) between certain ZFA-models and a certain class of groups.


Both of these applications of ZFA make use of so-called [permutation models](https://en.m.wikipedia.org/wiki/Permutation_model#CITEREFFraenkel1922).


I'd like to know what other applications ZFA might have. I would particularly be interested in applications that do not make use of permutation models.



Applications of ZFA-Set Theory
Although you prefer models other than permutation models, let me point out an appearance of permutation models, particularly the basic Fraenkel model, at the border between computer science and logic. The topic concerns operators that bind variables, like $\forall$ and $\exists$ in logic, the $\lambda$ in lambda-calculus, and $\int$ in calculus. The actual variables used with such an operator don't matter, i.e., bound variables can be renamed, as long as there are no clashes. That suggests thinking of variables as atoms in a ZFA universe, thinking of the renaming process as permuting the atoms, and requiring each formula to be invariant under enough permutations (all permutations fixing the formula's free variables). The theory built up from these ideas is called "nominal sets".


