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 $T_{\mathrm{a}\mathrm{s}}$ such that, given any sentence $\sigma$ in the language of pure metric spaces and any $ϵ>0$, the probability that the difference of the value of $\sigma$ in a random metric space of size $n$ and the value of $\sigma$ in any model of $T_{\mathrm{a}\mathrm{s}}$ is less than $ϵ$ approaches $1$ as $n$ approaches infinity. We also establish some model-theoretic properties of the theory $T_{\mathrm{a}\mathrm{s}}$.