The Urysohn sphere U is the unique separable metric space of diameter at most 1 with two important properties:  (1)  any separable metric space of diameter at most 1 embeds into U; (2)  any isometry between finite subspaces of $\mathfrak{U}$ extends to a self-isometry of U.  The Urysohn sphere is important both from a descriptive set-theoretic point of view and from a model-theoretic point of view as it can be viewed as the continuous analogue of either an infinite set or the random graph.

In this talk, I will present joint work with Bradd Hart showing that the Urysohn sphere is pseudofinite, meaning roughly that any first-order fact true in every finite metric space is also true in U.  Consequently, U satisfies an approximate 0-1 law which should be of independent combinatorial interest.  The proof uses an important fact from descriptive set theory and some basic probability theory.