  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
$n < \omega$, and also 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. We show that every degree c.e. in and above
$\mathbf{0}^{(\alpha)}$, for $\alpha$ a limit ordinal, is a degree of
categoricity. We also show that every degree c.e. in and above
$\mathbf{0}^{(\omega)}$ is the degree of categoricity of a prime model, making
progress towards a question of Bazhenov and Marchuk.

Logic