Let be a family of structures, closed under isomorphism, in a fixed computable language. We consider effective lists of structures from such that every structure in is isomorphic to exactly one structure on the list. Such a list is called a computable classification of , up to isomorphism. Using the technique of Friedberg enumeration, we show that there is a computable classification of the family of computable algebraic fields and that with a -oracle, we can obtain similar classifications of the families of computable equivalence structures and of computable finite-branching trees. However, there is no computable classification of the latter, nor of the family of computable torsion-free abelian groups of rank , even though these families are both closely allied with computable algebraic fields.
"Classifications of Computable Structures." Notre Dame J. Formal Logic 59 (1) 35 - 59, 2018. https://doi.org/10.1215/00294527-2017-0015