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$Σ_1$-Stationary logic as an $\aleph_1$-Abstract Elementary Class

Will Boney

Abstract

$μ$-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.

$Σ_1$-Stationary logic as an $\aleph_1$-Abstract Elementary Class

Abstract

-Abstract Elementary Classes are a model theoretic framework introduced in [BGL+16] to encompass classes axiomatized by . We show that the framework extends beyond these logics by showing classes axiomatized in with just the quantifier are an -Abstract Elementary Class.

Paper Structure

This paper contains 4 sections, 4 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Definitions and Notation
  3. Positive, Deliberate Axiomatizations
  4. The Example of Barwise-Kaufmann-Makkai

Key Result

Theorem 2.2

Fix $T \subset \mathbb{L}^{\Sigma_1}(aa)(\tau)$ and fix Then $\mathbb{K}_T^+$ is an $\omega_1$-Abstract Elementary Class with $LS(\mathbb{K}_T^+) =|T| + \aleph_1$.

Theorems & Definitions (9)

  • Definition 1.4: bglrv-muaecs
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.2
  • Example 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6