Who is lying? Wittgenstein or Gödel?

Abstract

Gödel’s Incompleteness Theorem was a major discovery in modern logic that has consistently attracted the attention of scientific and philosophical circles. However, since Gödel’s Incompleteness Theorem was put forward, the scientific and philosophical significance of its proof has been questioned, especially Wittgenstein, who strongly criticized it. Wittgenstein’s main argument against this theorem was that he believed Gödel sentence was a logical contradiction, that there was no such thing as a true and unprovable proposition at all, that it was a self-referential and self-contradictory concept, and that we should abandon the natural language explanation of this proposition. The author points out that Gödel sentence is a logically invalid formula constructed outside the domain of the principle of representability and lacks recursion; this contradiction stems from the abuse of formalization in language. At the same time, the author discovered a paradox in the proof of Gödel’s Incompleteness Theorem, that is, Gödel sentence itself is a kind of Russell’s Paradox. The author believes that the philosophical root of Russell’s Paradox and Gödel sentence lies in adhering to the idea of actual infinity, while the Infinite Exchange Paradox overturns the Cantor’s diagonal argument, which shakes the foundation of the idea of actual infinity. The above findings strongly support Wittgenstein’s view. Therefore, the conclusion of this paper is that Wittgenstein’s questioning of Gödel sentence is completely correct, the proof of Gödel’s Incompleteness Theorem is based on a logically invalid formula, and the contradiction of the proof originates from Gödel sentence itself, thus it cannot be attributed to the incompleteness of the system, so the proof is wrong.