Bounds on Scott Ranks of Some Polish Metric Spaces

TL;DR

This paper establishes upper bounds on the Scott ranks of certain Polish metric spaces, linking their complexity to the Church-Kleene ordinal of their dense subspaces, with implications for rigidity and computability.

Contribution

It provides new bounds on the Scott ranks of Polish metric spaces based on their dense subspaces and introduces bounds related to the Church-Kleene ordinal.

Findings

Scott rank of proper Polish spaces is at most _1^{\u03bc}+1

Scott rank of rigid Polish spaces is less than _1^{f

Bounds connect metric space complexity with computability theory

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 $\omega_1^{\mathcal{M}}$.