The almost sure theory of finite metric spaces

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