 John von Neumann’s (1925) untyped method for dealing with paradoxes, and with Russell’s paradox in particular, is simple and ingenious. Von Neumann introduces a distinction between membership and non-membership and, on this basis, draws a distinction between sets and classes. An object is a member (simpliciter) if it is a member of some class; and it is a non-member if it is not a member of any class. (Actually, von Neumann develops a theory of functions, taken as primitive, rather than classes, wherein corresponding to the member/non-member distinction one has a distinction between an object that can be an argument of some function and one that cannot. In its modern form, due to Bernays and  Gödel,  it is a single-sorted theory of classes.) 