Infinitary logics and abstract elementary classes
Saharon Shelah, Andrés Villaveces · Proceedings of the American Mathematical Society · 2021
We prove that every abstract elementary class (a.e.c.) with Löwenheim–Skolem–Tarski (LST) number κ \kappa and vocabulary τ \tau of cardinality ≤ κ \leq \kappa can be axiomatized in the logic L ℶ 2 ( κ ) + + + , κ + ( τ ) {\mathbb L}_{\beth _2(\kappa )^{+++},\kappa ^+}(\tau ) . An a.e.c. K \mathcal {K} in vocabulary τ \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 S = S K \mathcal S={\mathcal S_\mathcal {K}} of an a.e.c. K \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 λ 1 L^1_\lambda .