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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.

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  • $\begingroup$ Maybe I'm misunderstanding something. As far as I can tell, the main theorem is not that there are only a small number of potentiality classes, but rather potentiality classes of relations induced 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
    Commented Mar 3, 2015 at 19:03
  • $\begingroup$ From a model theory perspective, those are the only potentiality classes of interest (that are Borel) - the isomorphism relation for the set of countable models of an $L_{\omega_1,\omega}$-sentence is always such a class. The really interesting thing here would be if this showed there are only $\omega_1$ classes of Borel equivalence for such situations. I agree it sounds a bit dubious. $\endgroup$
    – Richard Rast
    Commented Mar 4, 2015 at 15:00
  • $\begingroup$ Fair enough. However, I think saying that these are the only interesting potentiality classes from perspective of model theory is a bit too general, Borel equivalence relations in model theory are not limited to isomorphism relations of countable models. $\endgroup$
    – tomasz
    Commented Mar 5, 2015 at 6:01
  • $\begingroup$ @RichardRast: If by "Borel equivalent" you mean "Borel bireducible", then the answer should be no. I recently had to quote this theorem, 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
    Commented Sep 7, 2015 at 6:50

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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 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 paper).

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