Categoricity Spectra for Rigid Structures

Abstract

For a computable structure $\mathcal{M}$, the categoricity spectrum is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable copies of $\mathcal{M}$. If the spectrum has a least degree, this degree is called the degree of categoricity of $\mathcal{M}$. In this paper we investigate spectra of categoricity for computable rigid structures. In particular, we give examples of rigid structures without degrees of categoricity.