[yongjik]

Okay, let me try to write these "algorithms" in a concrete term.

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

You are allowed to spend infinitely long time to return each digit. (That is, you can use constructions that are not even computable in finite time, as long as you can prove that given i and j, there is exactly one digit that satisfies your criterion.)

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

That's it.