             This argument shows that collections of sets of atoms need not necessarily have choice functions, but it fails to establish the same fact for the “usual” sets of mathematics, for example the set of real numbers. This had to wait until 1963 when Paul Cohen showed that it is consistent with the standard axioms of set theory (which preclude the existence of atoms) to assume that a countable collection of pairs of sets of real numbers fails to have a choice function. The core of Cohen’s method of proof—the celebrated method of forcing—was vastly more general than any previous technique; nevertheless his independence proof also made essential use of permutation and symmetry in essentially the form in which Fraenkel had originally employed them.