Classifying the complexity of models of arithmetic

TL;DR

This paper classifies the Scott complexities of models of Peano arithmetic by analyzing colored linear orderings and linking complexities to classical model-theoretic notions, offering new tools for understanding these models.

Contribution

It introduces a complete Scott analysis of colored linear orderings and connects Scott complexities with classical model-theoretic properties of Peano arithmetic models.

Findings

Constructed models of specific Scott complexities.

Linked Scott complexities with prime, finitely generated, and recursively saturated models.

Provided a comprehensive framework for analyzing models of Peano arithmetic.

Abstract

We classify the possible Scott complexities for models of Peano arithmetic. We construct models of particular complexities by first giving a complete Scott analysis of colored linear orderings and constructing models of Peano arithmetic from these colored orderings. We also provide tight connections of certain Scott complexities with notions from the classical theory of models of Peano arithmetic, such as prime, finitely generated, and recursively saturated. This effort provides a powerful set of tools to understand the models of Peano arithmetic.