> Without the decimal point these real numbers just become natural numbers.
No, they don't, because the vast majority of them have an infinite number of digits to the right of the decimal point. That's the key: there are more numbers with an infinite number of non-zero digits (the reals) than there are numbers with a finite number of non-zero digits (the naturals).
> The problem with Cantor's argument comes down to the fact that the procedure he uses to find a number not in the set is essentially the same as the procedure he uses for creating the infinite set in the first place.
Again, no. Cantor doesn't create the set, you do. The proof is like a game. It says: give me any procedure for (putatively) making a list of all of the real numbers, and I will give you back a number that is not in the list.
No, they don't, because the vast majority of them have an infinite number of digits to the right of the decimal point. That's the key: there are more numbers with an infinite number of non-zero digits (the reals) than there are numbers with a finite number of non-zero digits (the naturals).
> The problem with Cantor's argument comes down to the fact that the procedure he uses to find a number not in the set is essentially the same as the procedure he uses for creating the infinite set in the first place.
Again, no. Cantor doesn't create the set, you do. The proof is like a game. It says: give me any procedure for (putatively) making a list of all of the real numbers, and I will give you back a number that is not in the list.