Decidable or not

Question

Now define D , the diagonal set of strings:

$D=\left \{ w\epsilon \Sigma ^{*} \right \}$ where $w$  is not in $f\left ( w \right )$

• Call the correspondence $f$ is countable set $\Sigma ^{*}$
• For example 
• $f\left ( \varepsilon \right )=L_{0}$ ,i.e. NULL string
• $f\left ( 0 \right )=L_{1}$ , string only contain a $'0'$
• $f\left ( 1 \right )=L_{2}$  for all even length string of infinite language etc.

Now, the question is

1) is D countable?

2) is D decidable?

Answer 1

I think...