[lisper]

> my point is simply that there are two variants of this task

There are infinite variants of this task. But only one of them is mathematically interesting with respect to the claim that the reals cannot be put into one-to-one correspondence with the naturals.

> including waiting until after the supposed counter-example has been generated

Obviously, if I give you a real you can then generate a list that includes that real. That isn't very interesting.