  As stated above, in 1922 Fraenkel proved the independence of AC from a system of set theory containing “atoms”. Here by an atom is meant a pure individual, that is, an entity having no members and yet distinct from the empty set (so a fortiori an atom cannot be a set). In a system of set theory with atoms it is assumed that one is given an infinite set \(A\) of atoms. One can build a universe \(V(A)\) of sets over \(A\) by starting with \(A\), adding all the subsets of \(A\), adjoining all the subsets of the result, etc., and iterating transfinitely.  \(V(A)\) is then a model of set theory with atoms. The kernel of Fraenkel’s method for proving the independence of AC is the observation that, since atoms cannot be set-theoretically distinguished, any permutation of the set \(A\) of atoms induces a structure-preserving  permutation—an automorphism—of the universe \(V(A)\) of sets built from \(A\). This idea may be used to construct another model \(Sym(V)\) of  set theory—a permutation or symmetric model—in which a set of mutually disjoint pairs of elements of \(A\) has no choice function.  