The Scott rank of Polish metric spaces

Abstract

We study the usual notion of Scott rank but in the setting of Polish metric spaces. The signature consists of distance relations: for each rational $q>0$, there is a relation $R_{<q}(x,y)$ stating that the distance of $x$ and $y$ is less than $q$. We show that compact spaces have Scott rank at most $\omega$, and that there are discrete ultrametric spaces of arbitrarily high countable Scott rank.