Logic Seminar

In a 2012 research note, Fokina, Friedman, Koerwien 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 ω1. It was known, through work of Doucha, that the Scott rank can be at most ω1, but it was not known if rank ω1 could be attained. The proof that rank ω1 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 ω, iterated through the countable ordinals.