Degrees of Categoricity Above Limit Ordinals

Abstract

A computable structure $\mathcal{A}$ has degree of categoricity $\mathbf{d}$ if $\mathbf{d}$ is exactly the degree of difficulty of computing isomorphisms between isomorphic computable copies of $\mathcal{A}$. Fokina, Kalimullin, and Miller showed that every degree d.c.e. in and above $\mathbf{0}^{(n)}$, for any $n < ω$, and also the degree $\mathbf{0}^{(ω)}$, are degrees of categoricity. Later, Csima, Franklin, and Shore showed that every degree $\mathbf{0}^{(α)}$ for any computable ordinal $α$, and every degree d.c.e. in and above $\mathbf{0}^{(α)}$ for any successor ordinal $α$, is a degree of categoricity. We show that every degree c.e. in and above $\mathbf{0}^{(α)}$, for $α$ a limit ordinal, is a degree of categoricity. We also show that every degree c.e. in and above $\mathbf{0}^{(ω)}$ is the degree of categoricity of a prime model, making progress towards a question of Bazhenov and Marchuk.