$\Sigma_1$-Stationary logic as an $\aleph_1$-Abstract Elementary Class

TL;DR

This paper extends the framework of $eth$-Abstract Elementary Classes to include classes axiomatized in $ ext{L}(aa)$ logic with the $aa$ quantifier, broadening the scope of model-theoretic classes.

Contribution

It demonstrates that classes axiomatized in $ ext{L}(aa)$ with the $aa$ quantifier form an $eth_1$-Abstract Elementary Class, expanding the existing framework.

Findings

Classes axiomatized in $ ext{L}(aa)$ are $eth_1$-Abstract Elementary Classes.

The framework extends beyond $eth$-Abstract Elementary Classes.

The paper broadens the scope of model-theoretic classes.

Abstract

$\mu$-Abstract Elementary Classes are a model theoretic framework introduced in [BGL+16] to encompass classes axiomatized by $\mathbb{L}_{\infty, \infty}$. We show that the framework extends beyond these logics by showing classes axiomatized in $\mathbb{L}(aa)$ with just the $aa$ quantifier are an $\aleph_1$-Abstract Elementary Class.