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1 Bulletin of the Polish Academy of Sciences. Mathematics

1 2019

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Structural logic and abstract elementary classes with intersections

Boney Will, Vasey Sebastien

Bulletin of the Polish Academy of Sciences. Mathematics

2019

Vol. 67, no. 1

1--17

We give a syntactic characterization of abstract elementary classes (AECs) closed under intersections using a new logic with a quantifier for isomorphism types that we call structural logic: we prove that AECs with intersections correspond to classes of models of a universal theory in structural logic. This generalizes Tarski’s syntactic characterization of universal classes. As a corollary, we prove that any AEC closed under intersections with countable Löwenheim-Skolem-Tarski number is axiomatizable in L∞,ω(Q), where Q is the quantifier “there exist uncountably many”.