I provide an interpretation of Wittgenstein’s much criticised remarks on Gödel’s First Incompleteness Theorem in a paraconsistent framework: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was consequent upon 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. I show that the model-theoretic features of paraconsistent arithmetics match with many intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question.
|Title of host publication||Paraconsistency|
|Subtitle of host publication||Logic and Applications|
|Editors||K. Tanaka, F. Berto, E. Mares, F. Paoli|
|Number of pages||20|
|Publication status||Published - 2012|
|Name||Logic, Epistemology, and the Unity of Science|
The non-technical parts of this work draw on a paper published in Philosophia Mathematica, 17: 208–219, with the title “The Gödel Paradox and Wittgenstein’s Reasons”. I am grateful to Oxford University Press and to the Editors of Philosophia Mathematica for permission to reuse that material. I am also grateful to an anonymous referee for helpful comments on this expanded version