Infinitary Logics and Abstract Elementary Classes

Abstract

We prove that every abstract elementary class (a.e.c.) with LST number $\kappa$ and vocabulary $\tau$ of cardinality $\le \kappa$ can be axiomatized in the logic ${\mathbb{L}}_{{\beth }_2(\kappa {)}^{+++},{\kappa }^{+}}(\tau )$. 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 $L_{\lambda }^1$.}

14 pages