Infinitary logics and abstract elementary classes

Abstract:

We prove that every abstract elementary class (a.e.c.) with Löwenheim–Skolem–Tarski (LST) number $\kappa$ and vocabulary $\tau$ of cardinality $\leq \kappa$ can be axiomatized in the logic ${\mathbb L}_{\beth _2(\kappa )^{+++},\kappa ^+}(\tau )$. An a.e.c. $\mathcal {K}$ in vocabulary $\tau$ is therefore an EC class in this logic, 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 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 $L^1_\lambda$.