Bounds on Scott Ranks of Some Polish Metric Spaces

Abstract

If $\mathcal{N}$ is a proper Polish metric space and $\mathcal{M}$ is any countable dense submetric space of $\mathcal{N}$, then the Scott rank of $\mathcal{N}$ in the natural first order language of metric spaces is countable and in fact at most ${\omega }_1^{\mathcal{M}}+1$, where ${\omega }_1^{\mathcal{M}}$ is the Church-Kleene ordinal of $\mathcal{M}$ (construed as a subset of $\omega$) which is the least ordinal with no presentation on $\omega$ computable from $\mathcal{M}$. If $\mathcal{N}$ is a rigid Polish metric space and $\mathcal{M}$ is any countable dense submetric space, then the Scott rank of $\mathcal{N}$ is countable and in fact less than ${\omega }_1^{\mathcal{M}}$.