 Although it is the non-Hilbertian approach we will be concerned with in this entry, we briefly discuss the Hilbertian approach. The Hilbertian position differs because it depends on a distinction within mathematical language between a finitary sector, whose sentences express contentful propositions, and an ideal, or infinitary sector. Where exactly Hilbert drew the distinction, or where it should be drawn, is a matter of debate. Crucially, though, Hilbert adopted an instrumentalistic attitude towards the ideal sector. The formulae of this language are, or are treated as if they are, uninterpreted, having the syntactic form of sentences to which we can apply formal rules of transformation and inference but no semantics. Nonetheless they are, or can be useful, if the ideal sector conservatively extends the finitary, that is if no proof from finitary premisses to a finitary conclusion which takes a detour through the infinitary language yields a conclusion we could not have reached, albeit perhaps (herein lies the utility) by a longer, more unwieldy proof. The goal of the Hilbert programme was to provide a finitary proof of this conservative extension result; most, though not all, think this goal was proved impossible by Gödel’s second incompleteness theorem.