 Together, Wittgenstein’s finitism and his criterion of algorithmic decidability shed considerable light on his highly controversial remarks about putatively meaningful conjectures such as FLT and GC. GC is not a mathematical proposition because we do not know how to decide it, and if someone like G. H. Hardy says that he ‘believes’ GC is true (PG 381; LFM 123; PI §578), we must answer that s/he only “has a hunch about the possibilities of extension of the present system” (LFM 139)—that one can only believe such an expression is ‘correct’ if one knows how to prove it. The only sense in which GC (or FLT) can be proved is that it can “correspond to a proof by induction”, which means that the unproved inductive step (e.g., “\(G(n) \rightarrow G(n + 1)\)”) and the expression “\(\forall nG(n)\)” are not mathematical propositions because we have no algorithmic means of looking for an induction (PG 367). A “general proposition” is senseless prior to an inductive proof “because the question would only have made sense if a general method of decision had been known before the particular proof was discovered” (PG 402). Unproved ‘inductions’ or inductive steps are not meaningful propositions because the Law of the Excluded Middle does not hold in the sense that we do not know of a decision procedure by means of which we can prove or refute the expression (PG 400; WVC 82).