In a 2012 research note, Fokina, Friedman, Körwien and Nies (hereafter abbreviated FFKN) considered metric spaces as classical model-theoretic structures in a signature that has countable many binary relation used to express whether the distance between a given pair is less than or greater than a given rational. Then they considered the Scott analysis and Scott rank of metric spaces. One of the main questions asked by FFKN was if the Scott rank of a complete, separable metric space is countable.

In this talk, I will present an example of a complete, separable metric space which has Scott rank exactly \(\omega_1\). It was known, through work of Doucha, that the Scott rank can be at most \(\omega_1\), but it was not known if rank \(\omega_1\) could be attained. The proof that rank \(\omega_1\) is attained for the example given is interesting in that it uses a fair amount of moderately "serious" set theory. In particular, the proof relies on an idea which is similar in spirit to Stern's absoluteness and cardinality bounds for "virtual" Borel sets (i.e. Borel sets that exist in forcing extensions), and makes use of iterated powersets of \(\omega_1\), iterated through the countable ordinals.