The almost sure theory of finite metric spaces

TL;DR

This paper proves an approximate zero-one law for sentences in continuous logic over finite metric spaces, showing that large random spaces almost surely resemble a specific complete theory.

Contribution

It introduces a complete metric theory $T_{as}$ that characterizes the asymptotic behavior of sentences in finite metric spaces, bridging finite models and infinite theory.

Findings

Establishes an approximate zero-one law for continuous logic sentences.

Defines a complete theory $T_{as}$ capturing the asymptotic properties.

Shows convergence of sentence values in large finite metric spaces.

Abstract

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