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I am particularly interesting in the case where $T$ is o-minimal, but I would be interested in any theory $T$ (or even an $L_{\omega_1,\omega}$-sentence) which has this property.

Context: view the set of countable models of $T$ (under isomorphism) as an equivalence relation on a Polish space. Take a finite set of parameters $A$ (from some model of $T$) and then consider the set of countable models of $T_A$ (under $A$-isomorphism) as another equivalence relation on another Polish space. One could reasonably ask if the first is Borel-reducible to the second, or vice-versa.

In every example I know of, both are Borel-reducible to each other. However proving this in general has eluded me. I believe that in the general case (or even in the first-order-theory case) this is an open question. So I would like to know if this is the case, and either way, a link to any published results on this topic (or relevant examples either way).

Thank you!

Edit: the o-minimal case is no longer open (see https://arxiv.org/abs/1408.5876 ) but the more general case of this question is still open, as far as I know.

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    $\begingroup$ You say that the equivalence relation is "on a countable set", but is this really what you mean? I would think it lives instead on some uncountable Polish space... $\endgroup$
    – Joel David Hamkins
    Aug 23, 2013 at 18:53
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    $\begingroup$ Yes, that's my mistake. I meant the universes are countable, but of course the space of models is a Polish space (and typically uncountable). Editing now. $\endgroup$
    – Richard Rast
    Aug 23, 2013 at 18:57
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    $\begingroup$ I guess you can reduce to the case of adding just one constant, since either there is no change adding them one at a time, or else there is a change, which occurs when adding one more over what one has so far. $\endgroup$
    – Joel David Hamkins
    Aug 23, 2013 at 21:09

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