Degrees of Categoricity Above Limit Ordinals

TL;DR

This paper extends the understanding of degrees of categoricity in computable structures, showing that for limit ordinals, every c.e. degree above a certain level is a degree of categoricity, advancing the field's knowledge.

Contribution

It proves that every c.e. degree above $ extbf{0}^{(eta)}$ for limit ordinal $eta$ is a degree of categoricity, generalizing previous results to limit cases.

Findings

Every c.e. degree above $ extbf{0}^{(eta)}$ for limit ordinal $eta$ is a degree of categoricity.

Every degree c.e. in and above $ extbf{0}^{( extomega)}$ is the degree of categoricity of a prime model.

Progress towards classifying degrees of categoricity for all computable structures.

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 $n < \omega$, 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.