An argument concerning the unknowable

Abstract

I shall formulate and evaluate a new argument concerning the domain of the unknowable. It is an argument about (un)knowability. More specifically, it is an argument about what we can(not) 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 (as opposed to provability in an antecedently given formal system). Suppose, for a reductio, that there exists a property 0 of natural numbers such that it is provable that for some natural number n, 0(n) is true but unprovable. Then, by the least number principle, there must be a smallest such natural number n. Then there provably exists exactly one smallest number n such that 0(n) is true but unprovable. Then it is provable that the smallest n such that 0(n) is true but unprovable is true but unprovable. But then 0(n) is both provable and unprovable. But this is a contradiction. So there can be no property of natural numbers 0(x) such that it is provable that for some number «, 0(n) is true but unprovable. This argument is not specifically tied to the structure of the natural numbers. It is clear that a similar argument can be formulated for every mathematical domain for which we have a definable well-ordering. The structure of the (finite and transfinite) ordinal numbers constitutes one such domain. In the argument, no specific assumptions seem to be made about the property 0(x). If the description argument is valid, then its conclusion holds for purely arithmetical properties of natural numbers as well as for properties that also involve the notion of a priori knowability. Because reasoning about a definite description ('the smallest n such that 0(n) is true but unprovable') is a crucial component of the argumentation,