Which Classes of Structures Are Both Pseudo-elementary and Definable by an Infinitary Sentence?

TL;DR

This paper characterizes classes of structures that are both pseudo-elementary and definable by infinitary sentences without disjunctions, revealing their precise logical form and intersection.

Contribution

It identifies exactly which classes are both pseudo-elementary and $ ext{L}_{ ext{omega}_1 ext{omega}}$-elementary, showing they are definable by infinitary formulas without disjunctions.

Findings

Classes both pseudo-elementary and $ ext{L}_{ ext{omega}_1 ext{omega}}$-elementary are characterized.

Such classes are definable by infinitary formulas with no disjunctions.

The paper provides a precise logical description of the intersection.

Abstract

When classes of structures are not first-order definable, we might still try to find a nice description. There are two common ways for doing this. One is to expand the language, leading to notions of pseudo-elementary classes, and the other is to allow infinite conjuncts and disjuncts. In this paper we examine the intersection. Namely, we address the question: Which classes of structures are both pseudo-elementary and $\mathcal{L}_{\omega_1 \omega}$-elementary? We find that these are exactly the classes that can be defined by an infinitary formula that has no infinitary disjunctions.