Degrees of Categoricity for Prime and Homogeneous Models

Abstract

We study effective categoricity for homogeneous and prime models of a complete theory. For a computable structure \(\mathcal {S}\), the degree of categoricity of \(\mathcal {S}\) is the least Turing degree which can compute isomorphisms among arbitrary computable copies of \(\mathcal {S}\). We build new examples of degrees of categoricity for homogeneous models and for prime Heyting algebras, i.e. prime models of a complete extension of the theory of Heyting algebras. We show that \(\mathbf {0}^{(\omega +1)}\) is the degree of categoricity for a homogeneous model. We prove that any Turing degree which is d.c.e. in and above \(\mathbf {0}^{(n)}\), where \(3 \le n <\omega \), is the degree of categoricity for a prime Heyting algebra.