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== Borel classification questions ==

Let $T$ be a completion of $PA$. It is not hard to see that the isomorphism problem for finitely generated models of $T$, $\cong^{fg}_T$, is Borel.

Coskey and Kossak prove in ''The complexity of classification problems for models of arithmetic'', Bull. Symbolic Logic 16 (2010), no. 3, 345–358, that $\cong^{fg}_T$, is essentially countable and  $E_0\leq_B \cong^{fg}_T$ i.e. $\cong^{fg}_T$ is not  smooth. Is $\cong^{fg}_T$ hyperfinite? In other words, is $\cong^{fg}_T$ Borel reducible to $E_0$?