I will just diagonalize forever before I give the set to Cantor. If he claims that he can diagonalize the set to generate a new number not in the set then I will say "Oh, yeah - I missed one. I wasn't really done after all. Give it back to me and I will add some more to it before you get your hands on it!". As you can see, Cantor will never get a list because I will never really be done generating it.
I could just give him the list and say that whatever he adds is precisely what I would have added had I continued. This is like an infinite lazy list in programming.
I could just give him the list and say that whatever he adds is precisely what I would have added had I continued. This is like an infinite lazy list in programming.