Suppose that $\phi(x)$ is a $\Delta^{1}\_{2}$-formula (without parameters) and let $A:=\{x\subseteq\omega:\phi(x)\}$. It is clear that, e.g. if there are Cohen-generics over $L$, then $A$ cannot be the set of constructible reals, roughly because $\Delta^{1}\_{2}$ cannot distinguish between sufficiently high Cohen-generics in $L$ and Cohen-generics over $L$. I wonder whether this can be strengthenend in the following way:


Given appropriate largeness assumptions (existence of generic filters, large cardinals...), at least one of $A$ and $\mathfrak{P}(\omega)\setminus A$ contains real numbers of all degrees of constructibility. In other words, can $\Delta^{1}\_{2}$ "separate" degrees of constructibility in a nontrivial way?



Can $\Delta^{1}_{2}$ separate degrees of constructibility?
Expanding on Douglas Ulrich's answer:


Note that this question is only meaningful when there are nonconstructible reals since otherwise there is only one constructibility degree. 


If there is a nonconstructible real and $A$ is $\Delta\_2^1$ set, then either $A$ or $\mathbb{R} \setminus A$ is a $\Sigma\_2^1$ set of reals with a nonconstructible element. Without loss of generality, suppose $A$ has the nonconstructible real. By the Mansfield-Solovay theorem (using the $\omega\_1$-Suslin representation via the Shoenfield Tree $S$ for $A$ which belongs to $L$), there is a constructible tree $T$ so that $[T] \subseteq A$. 


Every constructible tree is an $L$-pointed tree. So $[T]$ has every $L$-degree. 


