I have been trying to solve an exercise from Kunen (1980) on Fraenkel and Mostowski's construction of a choiceless model of set theory. I have a couple of questions:
The model is constructed from an infinite set $U$ of "urelements", or "atoms", and it lacks a well-ordering of $U$ in it. If I understand correctly, not only does it lack a well-ordering of $U$ but it also lacks any linear orders on it. Am I correct?
In Kunen, instead of talking about urelements or atoms, one assumes ZF - foundation and working in that theory starts with a set $U$ that is infinite and whose elements are of the form $x = \{x\}$. It is another exercise in the book to show that the existence of such $U$ is consistent, but how can one do so? I know how to interpret the theory ZF - foundation + "such $U$ exists" by using graphs, but I would like to know how to obtain a (transitive) model that interprets $\in$ as $\in$. I find it hard to think about the latter possibility because for example under the presence of antifoundation (which is equiconsistent with the rest of the axioms) such $U$ has to be a singleton.