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 Lω1ω-elementary? We
find that these are exactly the classes that can be defined by an infinitary
formula that has no infinitary disjunctions