In Nies's monograph "Computability and Randomness", there is an interesting remark about $\Pi_2^0$-singletons, where he gives an argument that such singletons can be non-arithmetic and he gives the following proof:
There is a computable $ g $ such that $ (X^{[n]})' = W^X_{g(n)}$ for each $ X$ and $ n $, so $X = \emptyset^{(\omega)} \iff X^{[n]} = \emptyset \; \& \; \forall n, X^{[n+1]} = (X^{[n]})' \iff \forall n \, \forall k, s \, \exists t \geq s \left[ k \in X \leftrightarrow k \in W^{X_t}_{g(n),t} \right]$.
After the above proof, he goes on to shade some philosophical insights on the matter in that such an argument is an illustration of local vs global view of sets. I will add the paragraph in question for clarity:
Recall the discussion of the local versus the global view of sets in Section 1.3. If $Y \in Σ^0_n$ , the description showing this embodies the local view. This is even more apparent at the lower levels $Σ^1_0$ and $Σ^0_2$ of the arithmetical hierarchy, where we still have a reasonable effective approximation of Y . On the contrary, a description of a set Z as a $Π^0_n$ singleton, such as the description of $∅^{(ω)}$ above, embodies the global view. The description only works because it involves the set as a whole.
Now I know that we can think of a $\Pi^0_2$ class as a countable intersection of $\Sigma^0_1$ classes, which might qualify I guess as a local view of such a class.
My question is what would be a natural way of thinking about a local view of such a complicated $\Pi^0_2$ class?(I know that using an argument similar to that of Nies's above we can show that they go even further into the hyperarithmetic). I guess a way to look at it is that I know the generic, I just don't know a natural forcing notion for it.
