I would like to understand the complexity of "equality of Borel sets". By complexity, I mean the complexity in the sense of Borel reducibility.

Of course, since there is no standard Borel space of Borel sets of a Polish space, we have to work with *Borel codes* to make sense of "equality of Borel sets". Here is a nice paper by Clemens. In Section 1, he explains how we construct the space of Borel codes and what "equality of Borel sets" means (see Definition 8).

Let $E_B$ be the equivalence relation on Borel codes defined as follows: Two Borel codes are $E_B$-equivalent if and only if they define the same Borel set. It can be seen that the equivalence relation $E_B$ on the set of Borel codes is not a Borel equivalence relation (for example, see the remarks in Page 5). The set of Borel codes is not even a Borel set.

On the other hand, if we fix some $\alpha < \omega_1$, then the set ${BC}_{\alpha}$ of Borel codes of rank less than $\alpha$ is a Borel set. My question is whether or not $E_{\alpha}$, equivalence of Borel codes of rank less than $\alpha$, is Borel?

I specifically want to know if this is true when $\alpha$ is finite (more specifically, $\alpha=4$). My first reaction was to carry out an induction argument trying to bound $E_{\alpha}$ from above by using jumps or powers of the previous bounds but I can't seem to figure out what to do. I suspect that these relations might not be Borel even for small finite values.