Definability and Scott rank in separable Metric structures

TL;DR

This paper introduces a Scott rank concept for separable metric structures using continuous logic, extending classical ideas to the metric setting and exploring related definability and automorphism properties.

Contribution

It develops a continuous analogue of Scott rank for metric structures and proves new results on definability, type omission, and back-and-forth techniques.

Findings

Defined Scott rank for separable metric structures.

Proved results on definability and automorphism orbit closures.

Extended classical model theory concepts to continuous logic.

Abstract

We give a notion of Scott rank for separable metric structures based on the definability of the (metric closures of) automorphism orbits in continuous infinitary logic. This is a continuous analogue of work of Montalb\'an for countable structures. In the process, we prove some results concerning definability, type omitting, and back-and-forth for metric structures.