Regardless, I have a lot to ponder.
Yes; to be clear, my point is simply that there are two variants of this task:
- Given an integer i and our list (or the infinite procedure that generates it), then the task is easy since we can diagonalise.
- Given an integer i and no knowledge of our list, the task becomes impossible, since the supposed "real not in your list" is actually independent of the list (by definition); hence we're free to choose any list we like, including waiting until after the supposed counter-example has been generated, and sticking that at the head of our list.
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.
Assume that you have a way of generating a sequence R_1, R_2, R_3, ... of real numbers in [0, 1]. (For simplicity, let's just consider [0, 1], because it still has the same cardinalityas R.) You have your "algorithm":
generate_digit(i, j):
Somehow compute the j'th digit of R_i.
return the digit
Cantor's objective is to defeat your algorithm by generating a real number not in the list. Similarly as above, it can be written as a function that returns the j'th digit, given j. Behold it in its full glory:
defeat_indexing(j):
digit = generate_digit(j, j)
return (digit + 1) % 10