The almost sure theory of finite metric spaces

We establish an approximate zero‐one law for sentences of continuous logic over finite metric spaces of diameter at most 1. More precisely, we axiomatize a complete metric theory TAS such that, given any sentence σ in the language of pure metric spaces and any ε>0, the probability that the difference of the value of σ in a random metric space of size n and the value of σ in any model of TAS is less than ε approaches 1 as n approaches infinity. We also establish some model‐theoretic properties of the theory TAS.