I think this topic has been discussed before. In his personal notebooks, Luzin speculated that the Cantor diagonal proof shows only that the real numbers are not effectively enumerable. But I have always had trouble understanding what it could mean to say that " |X| =3D |N| " is true if no such enumeration "exists." But I think William Waterhouse's objection misses the point: the set X *is* the real numbers, by assumption. Of course taking Z in place of X produces a different result: so what? Or am I missing something myself?