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...

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Bibliographic Details
Published in:The bulletin of symbolic logic 2023-03, Vol.29 (1), p.1-18
Main Authors: BONEY, WILL, CSIMA, BARBARA F., DAY, NANCY A., HARRISON-TRAINOR, MATTHEW
Format: Article
Language:English
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Logic
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Summary: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.
ISSN:1079-8986
1943-5894
DOI:10.1017/bsl.2023.1