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, ≅fgT, is Borel.
Coskey and Kossak proved that ≅fgT, is essentially countable and E0≤B≅fgT i.e. ≅fgT is not smooth. Is ≅fgT hyperfinite? In other words, is ≅fgT Borel reducible to E0?
Reference: Coskey, Samuel, Kossak, Roman The complexity of classification problems for models of arithmetic, Bull. Symbolic Logic 16 (2010), no. 3, 345–358