Let S be the set of all computable sets (which are subsets of |N). We know that S is countable.

I have constructed the following proof that S is uncountable, using the diagonal argument, to which I wish to find the mistake.

Assume S is countable. Then it can be put in an ordering like this:

S1
S2
S3
...
Si
...

Let the predicate E_i(j) denote that the i-th element of S contains the natural number j; that is,
E_i is the characteristic function of S_i.

Now, construct the set S' such that its characteristic function E(x) = not E_x(x).

S' is computable, because the following algorithm can be constructed:

    function S'(i) {
        return !E_i(i);
    }

As E_i(i) can be computed by calling program S_i with input i in finite time, then therefore S'(i)
must also complete in finite time.

For all i in |N, E will be different from E_i, because they differ at element i. Therefore, the
set of computable numbers is uncountable.