Infinitary Logics and A.E.C

Saharon Shelah, Andrés Villaveces

Published 2020-10-05Version 1

We prove that every a.e.c. with LST number $\leq \kappa$ and vocabulary $\tau$ of cardinality $\leq \kappa$ can be defined 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 $L^1_\lambda$.