Russell’s Paradox, Gödel’s Theorem

Abstract

Smullyan has closely examined Gödel’s proof of arithmetic incompleteness both abstractly, for a mathematically sophisticated audience, and via the medium of puzzles, for a general audience. In this essentially pedagogical paper I take an intermediate position, and show that Gödel’s ideas fit naturally into the context of the elementary set theory that any mathematics student would know near the beginning of studies. Gödel once wrote, about his incompleteness proof, “The analogy of this argument with the Richard antinomy leaps to the eye. It is closely related to the ‘Liar’ too.” And further, “Any epistemological antinomy could be used for a similar proof of the existence of undecidable propositions.” Here, combining the spirit of both Smullyan’s formal and his popular work, I show how Russell’s paradox can be used in a direct and natural way to obtain Gödel’s incompleteness results.