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Warp said:My thinking is (and where I'm probably mistaken, although I don't know the details) that if we assume the set is countable, ie. enumerable, it shouldn't make any difference if we replace every element in the list with a natural number. From the perspective of the proof it should make no difference. (Or should it?)
That makes a big difference. Then you simply have a list of the natural numbers. And all you are showing then is that there exist real numbers that aren't natural numbers.