Structural Logic and Abstract Elementary Classes with Intersection

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 obtain that any AEC with countable Löwenheim-Skolem number is axiomatizable in ${\mathbb{L}}_{\infty ,\omega }(Q)$, where $Q$ is the quantifier "there exists uncountably many".

14 pages

References in corpus (3)

• Shelah's eventual categoricity conjecture in universal classes: part I
• Shelah's eventual categoricity conjecture in universal classes. Part II
• Universal abstract elementary classes and locally multipresentable categories

Cited by in corpus (2)

• On categoricity in successive cardinals
• Cofinality quantifiers in Abstract Elementary Classes and beyond