| Résume | Lindstrom’s theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Lowenheim Skolem Theorem. It has been one of the main tasks of Infinitary Model Theory to find stronger logics with similar characterizations with the hope that they would have useful applications. Despite intensive efforts no such logic was found until Shelah introduced his logic L^1_\kappa. Unfortunately, by that time most of the researchers working on the problem have left the subject. We try to revive the subject by introducing a class of logics which give Shelah’s logic in the limit and test their expressive power. We also give an alternative characterization of Shelah’s logic in terms of the Lowenheim-Skolem-Union property. This is joint work with J. Vaananen |
