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Philosophia Mathematica Advance Access originally published online on February 3, 2009
Philosophia Mathematica 2009 17(2):208-219; doi:10.1093/philmat/nkp001
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© The Author [2009]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

The Gödel Paradox and Wittgenstein's Reasons

Francesco Berto*

* IHPST, Sorbonne-École Normale Supérieure, 13 rue du Four, 75006 Paris, France. bertofra{at}unive.it

An interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match with some intuitions underlying Wittgenstein's philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question.


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