Infinitary Logics and Abstract Elementary Classes
Saharon Shelah, Andrés Villaveces
TL;DR
The paper proves that every abstract elementary class (a.e.c.) with LST number $\kappa$ and vocabulary of size $\leq \kappa$ is axiomatizable by a single infinitary sentence in ${\mathbb L}_{\beth_2(\kappa)^{+++},\kappa^+}(\tau)$, elevating the a.e.c. to an EC class. Central to the construction is the canonical tree $\mathcal{S}_{\mathcal K}$ of κ-sized models, which encodes all embeddings and enables the definition of a hierarchy of formulas $\varphi_{M,\gamma, n}$ that approximate the class. The main results include the existence of a sentence $\psi_{\mathcal K}$ with $\mathcal K=Mod(\psi_{\mathcal K})$ and a syntactic Tarski–Vaught‑like criterion for $\prec_{\cal K}$, together with a detailed analysis of the definability and connections to related logics such as $L^1_\kappa$. These advances substantially sharpen the definability of a.e.c.'s and clarify the landscape of infinitary logics that capture them.
Abstract
We prove that every abstract elementary class (a.e.c.) with LST number $κ$ and vocabulary $τ$ of cardinality $\leq κ$ can be axiomatized in the logic ${\mathbb L}_{\beth_2(κ)^{+++},κ^+}(τ)$. In this logic an a.e.c. is therefore an EC class rather than merely a PC class. This constitutes a major improvement on the level of definability previously given by the Presentation Theorem. As part of our proof, we define the \emph{canonical tree} $\mathcal S={\mathcal S}_{\mathcal K}$ of an a.e.c. $\mathcal K$. This turns out to be an interesting combinatorial object of the class, beyond the aim of our theorem. Furthermore, we study a connection between the sentences defining an a.e.c. and the relatively new infinitary logic $L^1_λ$.}