  We next define the universe \(Sym(V)\) to consist of the hereditarily symmetric members of \(V(A)\), that is, those \(x\in V(A)\) such that \(x\), the elements of \(x\), the elements of elements of \(x\), etc., are all symmetric. \(Sym(V)\) is also a model of set theory with set of atoms \(A\), and \(\pi\) induces an automorphism of \(Sym(V)\).