# Complexity and classification of countable models

From Peano's Parlour

## 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 proved 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 $E_0$?

Reference: Coskey, Samuel, Kossak, Roman *The complexity of classification problems for models of arithmetic*, Bull. Symbolic Logic 16 (2010), no. 3, 345–358