Abstract

Williamson has forcefully argued that Fitch's argument shows that the domain of the unknowable is non-empty. And he exhorts us to make more inroads into the land of the unknowable. Concluding his discussion of Fitch's argument, he writes: " Once we acknowledge that [the domain of the unknowable] is non-empty, we can explore more effectively its extent. … We are only beginning to understand the deeper limits of our knowledge. " I shall formulate and evaluate a new argument concerning the domain of the unknowable. It is an argument about knowability. More specifically, it is an argument about what we can know about the natural numbers. Since the domain of discourse will be the natural numbers structure, the notion of knowability can for the purposes of the argument be identified with a priori knowability or – which amounts to the same thing – absolute provability .Suppose, for a reductio, that there exists a property θ of natural numbers such that it is provable that for some natural number n, θ is true but unprovable. …