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 all degrees d.c.e. in and above $\mathbf{0}^{(n)}$ for any $n < \omega$ – as well as the degree $\mathbf{0}^{(\omega)}$ – are degrees of categoricity. Later, Csima, Franklin, and Shore showed that every degree $\mathbf{0}^{(\alpha)}$ for any computable ordinal $\alpha$, and every degree d.c.e. in and above $\mathbf{0}^{(\alpha)}$ for any successor ordinal $\alpha$, is a degree of categoricity. In this talk, I will show that for any limit ordinal $\alpha$, every degree c.e. in and above $\mathbf{0}^{(\alpha)}$ is a degree of categoricity.

This is joint work with Barbara Csima, Matthew Harrison-Trainor, and Mohammad Assem Mahmoud.