The diagonal argument Cantor uses to prove the uncountability of the reals cannot be applied to the naturals. That’s because every natural number has but finitely many digits. Consider this list of a few natural numbers:

1
2
3
4

You can change the first digit of the first number in that list, but the second number doesn’t even have a second digit, so there’s nothing to change.

Besides, the definition of countabilty of a set S does not mean that a person or device can count the elements of S (rattle off the sequence “1, 2, 3, …” while pointing out successive members of S) in a finite time, it requires only that it be possible to associate each element of S with a unique natural number. That’s what’s at the heart of @ThomasAndrews’ comment, because one can simply associate each natural number with itself. Cantor’s argument shows that any way you associate each natural number to a real, there will always be a real that’s left out, that doesn’t receive any natural number.

The diagonal argument Cantor uses to prove the uncountability of the reals cannot be applied to the naturals. That’s because every natural number has but finitely many digits, but the thing produced by the diagonal approach has infinitely many digits, so it cannot be a natural number (let alone a natural number missing from the list).

Besides, the definition of countabilty of a set S does not mean that a person or device can count the elements of S (rattle off the sequence “1, 2, 3, …” while pointing out successive members of S) in a finite time, it requires only that it be possible to associate each element of S with a unique natural number. That’s what’s at the heart of @ThomasAndrews’ comment, because one can simply associate each natural number with itself. Cantor’s argument shows that any way you associate each natural number to a real, there will always be a real that’s left out, that doesn’t receive any natural number.