We study effective categoricity of computable abelian groups of the form i∈ω H, where H is a subgroup of (Q, +). Such groups are called homogeneous completely decomposable. It is well-known that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite. We study ∆ 0 n-categoricity in this class of groups, for n… (More)
We show that the index set complexity of the computably categorical structures is Π 1 1-complete, demonstrating that computable categoricity has no simple syntactic characterization. As a consequence of our proof, we exhibit, for every computable ordinal α, a computable structure that is computably categorical but not relatively ∆ 0 α-categorical.