I will prove the recent result by David Kueker, that finitary AEC are closed under L_infinity_omega equivalence. This implies several definability results for finitary abstract elementary classes and also raises some open questions.
The proof does not use advanced model theory, but countably closed and unbounded sets and the cub-game, which was introduced by Kueker in 1977. The proof resembles the interplay between cub sets and the axioms for abstract elementary classes with countable Löwenheim-Skolem number. We also explain the role of the property finite character.