Structural Logic and Abstract Elementary Classes with Intersection

TL;DR

This paper introduces a new logical framework called structural logic to syntactically characterize abstract elementary classes with intersections, extending classical results and showing axiomatizability in an enriched infinitary logic.

Contribution

It provides a novel syntactic characterization of AECs with intersections using structural logic, generalizing Tarski's universal class characterization.

Findings

AECs with intersections correspond to models of universal theories in structural logic

Any AEC with countable Löwenheim-Skolem number is axiomatizable in _{,}(Q)

Structural logic includes a quantifier for isomorphism types

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 L\"owenheim-Skolem number is axiomatizable in $\mathbb{L}_{\infty, \omega} (Q)$, where $Q$ is the quantifier "there exists uncountably many".