The Classification Problem for Compact Computable Metric Spaces

We adjust methods of computable model theory to effective analysis. We use index sets and infinitary logic to obtain classificationtype results for compact computable metric spaces. We show that every compact computable metric space can be uniquely described, up to an isomorphism, by a computable Π3 formula, and that orbits of elements are uniformly given by computable Π2 formulas. We show that the index set for such spaces is Π3-complete, and the isomorphism problem is Π2-complete within its index set. We also give further classification results for special classes of compact spaces, and for other related classes of Polish spaces. Finally, as our main result we show that each compact computable metric space is ∆3-categorical, and there exists a compact computable Polish space which is not ∆2-categorical.