A Closer Look at the Russell Paradox

Abstract

Two types of approximation to the paradoxical Russell Set are presented, one approximating it from below, one from above. It is shown that any lower approximation gives rise to a better approximation containing it, and that any upper approximation contains a distinct better approximation. The Russell Paradox is then seen to be the claim that two of these processes of better approximations stop, and at the same set. This suggests that the unrestricted Axiom of Comprehension is, not a coherent intuition worthy of rescue from a mysterious paradox, but simply wishful thinking, a confusion of sets as extensional objects with classes defined by a property.