Infinitary stability theory

We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois (orbital) type of length less than a fixed cardinal kappa. We show: Theorem 0.1 (The semantic-syntactic correspondence) AnAEC K is fully (= LS(K). (2) K does not have the order property (defined in terms of Galois types). (3) There exist cardinals mu and lambda(0) with mu <= lambda(0) < beth((2LS(K))) + such that K is Galois stable in any lambda >= lambda(0) with lambda = lambda( LS(K). If K is Galois stable, then the class of kappa-Galois saturated models of K admits an independence notion ((