A lecture for the CUNY Graduate Center’s Model Theory Seminar.

Abstract: The main recent logic I will describe is Shelah’s infinitary logic (from 2012). I will describe some of the reasons for studying this logic (roughly, it is an infinitary logic that has interpolation and a weak form of compactness – therefore particularly well-adapted to model theory, as well as closure under chains) and some of the features lacking (mostly, a workable syntax). I will describe two other logics that have been created in order to capture better the syntax (one of these logics is my joint work with Väänänen, the other one is due originally to Karp and Cunningham and has recently been connected to by Dzamonja and Väänänen. Finally I will connect these logics with the problem of axiomatizing abstract elementary classes. In particular, I will describe *canonical trees* of models that enables one to build a sentence to test models for membership into aecs. This last part is joint work with Shelah.