Metric Spaces Are Universal for Bi-interpretation with Metric Structures

TL;DR

This paper demonstrates that metric spaces can serve as universal structures for bi-interpretation of metric structures, providing an explicit encoding that parallels classical discrete structure encodings but with unique topological considerations.

Contribution

It introduces an explicit encoding of metric structures into pure metric spaces, establishing their universality for bi-interpretation within the framework of metric model theory.

Findings

Metric structures can be encoded as pure metric spaces.

Such encodings are universal for bi-interpretation among metric structures.

The work reveals topological nuances absent in discrete settings.

Abstract

In the context of metric structures introduced by Ben Yaacov, Berenstein, Henson, and Usvyatsov, we exhibit an explicit encoding of metric structures in countable signatures as pure metric spaces in the empty signature, showing that such structures are universal for bi-interpretation among metric structures with positive diameter. This is analogous to the classical encoding of arbitrary discrete structures in finite signatures as graphs, but is stronger in certain ways and weaker in others. There are also certain fine grained topological concerns with no analog in the discrete setting.