A Closer Look at the Russell Paradox

TL;DR

This paper analyzes the Russell Paradox through approximations from below and above, revealing that the paradox arises from a mistaken belief that these approximation processes should terminate at the same set, challenging the coherence of the Axiom of Comprehension.

Contribution

It introduces a novel approximation framework for the Russell Paradox, clarifying its nature and questioning the validity of the unrestricted Axiom of Comprehension.

Findings

Lower approximations lead to better, larger sets containing the original.

Upper approximations contain distinct, better approximations.

The paradox arises from expecting these processes to converge at the same set.

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.