Approximate Categoricity in Continuous Logic

TL;DR

This paper extends the concept of approximate categoricity in continuous logic using distortion systems, generalizing previous frameworks and exploring analogs of Morley's theorem for uncountable models.

Contribution

It generalizes Ben Yaacov's characterization of approximate categoricity from perturbation to distortion systems and advances the understanding of Morley's theorem analogs in this context.

Findings

Extended Ryll-Nardzewski characterization to distortion systems

Progress towards Morley's theorem for inseparable approximate categoricity

Examples illustrating the interaction between separable and inseparable categoricity

Abstract

We explore approximate categoricity in the context of distortion systems, introduced in our previous paper, which are a mild generalization of perturbation systems, introduced by Ben Yaacov. We extend Ben Yaacov's Ryll-Nardzewski style characterization of separably approximately categorical theories from the context of perturbation systems to that of distortion systems. We also make progress towards an analog of Morley's theorem for inseparable approximate categoricity, showing that if there is some uncountable cardinal $\kappa$ such that every model of size $\kappa$ is 'approximately saturated,' in the appropriate sense, then the same is true for all uncountable cardinalities. Finally we present some examples of these phenomena and highlight an apparent interaction between ordinary separable categoricity and inseparable approximate categoricity.