Finite computable dimension and degrees of categoricity

Abstract

We first give an example of a rigid structure of computable dimension 2 such that the unique isomorphism between two non-computably isomorphic computable copies has Turing degree strictly below 0″, and not above 0′. This gives a first example of a computable structure with a degree of categoricity that does not belong to an interval of the form [0(α),0(α+1)] for any computable ordinal α. We then extend the technique to produce a rigid structure of computable dimension 3 such that if d0, d1, and d2 are the degrees of isomorphisms between distinct representatives of the three computable equivalence classes, then each di<d0⊕d1⊕d2. The resulting structure is an example of a structure that has a degree of categoricity, but not strongly.