Axiomatizing AECs and applications

TL;DR

This paper develops axiomatization results for abstract elementary classes (AECs), extending prior work by providing new bounds, simplifying existing theorems, and applying these to broader classes including $$-AECs.

Contribution

It introduces new axiomatization theorems for AECs with improved bounds and extends Shelah's presentation theorem, also applying these results to broader classes.

Findings

AECs have axiomatizations in extended logics with game quantification.

Under certain conditions, AECs are axiomatizable in $L_{eth_ heta^+, heta^+}$.

Improved bounds on the number of types in Shelah's presentation theorem.

Abstract

For any abstract elementary class (AEC) ${\bf K}$ with $\lambda=LS({\bf K})$, the following holds: 1. $K$ has an axiomatization in $L_{(2^\lambda)^+,\lambda^+}$, allowing game quantification. If ${\bf K}$ has arbitrarily large models, the $\lambda$-amalgamation property and is categorical both in $\lambda$ and $\lambda^+$, then it has an axiomatization in $L_{\lambda^{+},\lambda^{+}}$ with game quantification. These extend Kueker's result which assumes finite character and $\lambda=\aleph_0$. 2. If $K$ is universal and categorical in $\lambda$, then it is axiomatizable in $L_{\lambda^+,\lambda^+}$. 3. Shelah's celebrated presentation theorem asserts that for any AEC ${\bf K}$ there is a first-order theory in an expansion of $L({\bf K})$, and a set $\Gamma$ of $2^\lambda$ many $T$-types such that $K=PC(T,\Gamma,L({\bf K}))$. We provide a better bound on $|\Gamma|$ in terms of $I_2(\lambda,{\bf K})$. 4. We present additional applications which extend, simplify and generalize results of Shelah and Shelah-Vasey. Some of our main results generalize to $\mu$-AECs.