What is the flaw in the proof of the countability of the set of finite language?

Question

Let FT be the class of all finite languages over {0,1}.

Suppose FL is countable, so we can enumerate all members of FL as

FL = {L_0, L_1, L_2, ...}

Meanwhile, we also enumerate all strings over the alphabet as

\Sigma^*={w_0, w_1, w_2, ...}

Now consider the diagonal language

D={w_i: w_i is not in L_i, i=0,1,2,...}

which should be a member of FL.

However, for any i, D is not equal to L_i and we have reached a contradiction.

So FT is not countable.