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

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 $\mathcal{L}_{ω_1 ω}$-elementary? We find that these are exactly the classes that can be defined by an infinitary formula that has no infinitary disjunctions.

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