INFINITARY LOGICS and ABSTRACT ELEMENTARY CLASSES

We prove that every abstract elementary class (a.e.c.) with Löwenheim–Skolem–Tarski (LST) number κ and vocabulary τ of cardinality ≤ κ can be axiomatized in the logic L₂(κ₎+++_,κ+(τ). An a.e.c. K in vocabulary τ 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 S = S_K of an a.e.c. 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.