Since long ago it is known the existence of non-recursive sets (i.e., non-$\Delta_1$), e.g., Halting problem. It is also known (firstly noticed by Trakhtenbrot, and deeply studied by Smullyan) the existence of (non-trivial) pairs of sets that are *recursively inseparable*; this means that there are sets $A,B \subseteq \mathbb{N}$ such that $A \subsetneq B$, and there is no recursive set $C$ such that $A \subseteq C \subseteq B$. An example of pair with this property is to consider $A$ as the set of (Gödel numbers of) first-order valid formulas, and $B$ as the set of (Gödel numbers of) first-order formulas valid in all finite structures.

I am interested on what it is known about the "natural" generalization of the provious notion to $\Delta_2$sets, and to $\Delta_3$, etc. To be more precise: what is it known about pairs of sets $A,B \subseteq \mathbb{N}$ such that:

- $A \subsetneq B$,
- there is no $\Delta_2$ set $C$ such that $A \subseteq C \subseteq B$.

Is there any such pair? Is there a general theory for this notion as the one developed by Smullyan for $\Delta_1$? Has this been studied in some papers?

*Update 1*: I am interested in non trivial and explicit examples. For the non trivial part it makes more sense to replace $A \subsetneq B$ with the following conditions: $A \subseteq B$, and $B \setminus A$ is infinite.

*Update 2*: To make my question closer to what happens in the $\Delta_1$ example given above let me add the constraints that $A$ is $\Sigma_2$-complete and B is $\Pi_2$-complete.