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The set of numbers which can be written with finitely many members of a finite set of symbols is countable, sure.

That doesn’t sound like the claim the author is making in the sentence I quoted but maybe I am being uncharitable?

If you allow infinitely long descriptions, then I don't know what's the score. In the case of "descriptions" that amount to (potentially infinite) decimal expansions, I think an argument akin to Cantor's diagonal will show that you can't possibly describe all real numbers (or all numbers between 0 and 1), but I'm not confident I can assert this to be true.

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