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Structural logic and abstract elementary classes with intersections

Volume 67 / 2019

Will Boney, Sebastien Vasey Bulletin Polish Acad. Sci. Math. 67 (2019), 1-17 MSC: Primary 03C48; Secondary 03B60, 03C80, 03C95. DOI: 10.4064/ba8178-12-2018 Published online: 4 February 2019

Abstract

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 $\mathbb {L}_{\infty , \omega } (Q)$, where $Q$ is the quantifier “there exist uncountably many”.

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