About “The Philosophical Significance of Gödel’s Theorem”: Some Issues

Abstract

It is a very natural supposition that, for any particular consistent formal system of arithmetic, one of the pair consisting of the Gödel sentence and its negation must be true. This was rejected by Wittgenstein in the notorious appendix on Gödel’s Theorem in the Remarks on the Foundations of Mathematics.¹ Wittgenstein there implicitly repudiated not merely any Platonist conception of mathematics, as usually conceived, but the much more deeply rooted idea that arithmetic is in the business of description of a proper subject matter of any kind. His view, it seems, was that there simply is no defensible conception of truth for the sentences of a formal arithmetic which might coherently whether or not justifiably — be thought to outrun derivability within it.