The Gödel Paradox and Wittgenstein’s Reasons 1. The Implausibile Wittgenstein Wittgenstein’s notorious comments on Gödel’s First Incompleteness Theorem in the

Remarks on the Foundations of Mathematics were dismissed by early commentators, such as Kreisel, Anderson, Dummett, and Bernays, as an unfortunate episode in the career of a great philosopher. It appears that Wittgenstein had in his sights only the informal account of the Theorem, presented by Gödel in the introduction of his celebrated 1931 paper, and was misguided by it (not that he was the only one: because of the misunderstandings it originated, Helmer said that that exposition “without any claim to complete precision”1 is the only mistake in Gödel’s paper). It is claimed that Wittgenstein erroneously considered essential the natural language interpretation of the Gödel sentence, whose undecidability within (the modified system considered by Gödel, taken from) Russell and Whitehead’s Principia mathematica is at the core of the First Theorem, as claiming “I am not provable”. On the contrary, Gödel’s proof can be framed in syntactic terms in which no extra-mathematical interpretation of the formulas is required. Commentators were particularly struck by the fact that Wittgenstein seems to take the Gödel formula as a paradoxical sentence, not too different from the usual Liar – and Gödel’s proof itself, therefore, as the deduction of an inconsistency: 11. Let us suppose I prove the unprovability (in Russell’s system) of P; then by this proof I have proved P. Now if this proof were one in Russell’s system – I should in this case have proved at once that it belonged and did not belong to Russell’s system. – That is what comes of making up suc