 As with his intermediate views on genuine irrationals and the Multiplicative Axiom, Wittgenstein here looks at the diagonal proof of the non-denumerability of “the set of transcendental numbers” as one that shows only that transcendental numbers cannot be recursively enumerated. It is nonsense, he says, to go from the warranted conclusion that these numbers are not, in principle, enumerable to the conclusion that the set of transcendental numbers is greater in cardinality than the set of algebraic numbers, which is recursively enumerable. What we have here are two very different conceptions of a number-type. In the case of algebraic numbers, we have a decision procedure for determining of any given number whether or not it is algebraic, and we have a method of enumerating the algebraic numbers such that we can see that ‘each’ algebraic number “will be” enumerated. In the case of transcendental numbers, on the other hand, we have proofs that some numbers are transcendental (i.e., non-algebraic), and we have a proof that we cannot recursively enumerate each and every thing we would call a “transcendental number”.