An introduction to the Scott complexity of countable structures and a survey of recent results

TL;DR

This paper introduces the Scott complexity of countable structures, explaining how Scott sentences characterize structures up to isomorphism, and surveys recent progress in understanding their complexity.

Contribution

It provides an accessible introduction to Scott complexity and summarizes recent research developments in the field.

Findings

Characterization of countable structures via Scott sentences

Measurement of the complexity of describing structures

Summary of recent advances in Scott complexity

Abstract

Every countable structure has a sentence of the infinitary logic $\mathcal{L}_{\omega_1 \omega}$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs where possible, followed by a survey of recent advances.