The Urysohn sphere is pseudofinite

TL;DR

This paper proves that the Urysohn sphere is pseudofinite and establishes an approximate 0-1 law for finite metric spaces with diameter at most 1, linking model theory and metric geometry.

Contribution

It demonstrates the pseudofiniteness of the Urysohn sphere and derives an approximate 0-1 law for certain finite metric spaces, a novel connection in metric model theory.

Findings

Urysohn sphere is pseudofinite

Established an approximate 0-1 law for finite metric spaces of diameter ≤ 1

Bridged concepts between model theory and metric geometry

Abstract

We show that the Urysohn sphere is pseudofinite. As a consequence, we derive an approximate $0$-$1$ law for finite metric spaces of diameter at most $1$.