In a 1998 paper by Hjorth, Kechris, and Louveau, there was a definition given of a "potentiality class." That is, given an invariant equivalence relation $E$ on a standard Borel space $X$, we say $E$ is (for example) *potentially $\Pi^0_3$* if, for some Polish topology on $X$ yielding the same Borel sets, $E$ is a $\Pi^0_3$ subset of $X\times X$. This condition is equivalent to being Borel reducible to some $F$ which is actually $\Pi^0_3$ on its space, so these potentiality classes are downward closed with respect to Borel reducibility.

We say $E$ *is* $Pot(\Gamma)$ (for any such $\Gamma$) if it is potentially in $\Gamma$, but not potentially in the dual space (or, if $\Gamma$ is self-dual, if it is potentially in $\Gamma$ but not in any proper sub-class).

The main theorem is that there are only a small number of potentiality classes. What they do not directly say, but perhaps assume is obvious, is that if two relations are in the same potentiality class, then they *are* Borel equivalent. Is this a correct reading of the condition? If not, I'm not sure why not, but if so, it has a lot of interesting model-theoretic consequences and I would think they would point it out explicitly.

The paper in question is "Borel equivalence relations induced by actions of the symmetric group" by Hjorth, Kechris, Louveau, in the Annals of Pure and Applied Logic, 92 (1998) 63-112.

notthat there are only a small number of potentiality classes, but rather potentiality classes of relationsinduced by closed subgroups of $S_\infty$! This seems like quite a difference. I don't know about this relation, but in general, there are no more than $\omega_1$ potentiality classes, while there are $\mathfrak c$ many classes of Borel equivalence, so without CH at the very least, the two can't be the same (and even with CH it sounds dubious). $\endgroup$ – tomasz Mar 3 '15 at 19:03thistheorem, which should be somewhere in the paper you linked (but I did not bother checking). Given this fact, every essentially countable Borel equivalence relation which is the orbit equivalence relation of a Borel action of $S_{\infty}$ is potentially$\Sigma^0_2$. You can easily find three non-equivalent such orbit equivalence relations. (For example, isomorphism of torsion-free abelian groups of rank 1,2,3) $\endgroup$ – Burak Sep 7 '15 at 6:50