A presentation theorem for continuous logic and Metric Abstract Elementary Classes

TL;DR

This paper establishes a presentation theorem linking continuous logic and Metric Abstract Elementary Classes with classical logical frameworks, analyzing dense subsets and extending to types and saturation.

Contribution

It introduces a novel presentation theorem connecting continuous logic and Metric Abstract Elementary Classes to classical frameworks via dense subsets and closure properties.

Findings

Established a correspondence between continuous logic and classical frameworks.

Extended the presentation to types and saturation.

Provided a method to analyze dense subsets closed under functions.

Abstract

We give a presentation theorem for continuous first-order logic and Metric Abstract Elementary classes in terms of $L_{\omega_1, \omega}$ and Abstract Elementary Classes, respectively. This presentation is accomplished by analyzing dense subsets that are closed under functions. We extend this correspondence to types and saturation.