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.



Potentiality classes and Borel reductions
It is not necessarily true that Borel equivalence relations that are potentially in the same pointclass are Borel bireducible. For example, consider the orbit equivalence relations of the logic action of $S\_{\infty}$ on the standard Borel space of torsion-free abelian groups of rank $n$. Then these orbit equivalence relations are essentially countable and hence are potentially $\mathbf{\Sigma^0\_2}$ (for example, see [this](https://mathoverflow.net/questions/215709/natural-examples-of-bf-sigma0-3-equivalence-relations/215743#215743) theorem). However, Simon Thomas proved that the Borel complexity of isomorphism of torsion-free abelian groups of rank $n$ (strictly) increases with rank (in [this](http://www.ams.org/journals/jams/2003-16-01/S0894-0347-02-00409-5/) paper).


