Is the continuum hypothesis still open? If we interpret it as finding the laws of cardinal arithmetic (really exponentiation since addition and multiplication were classically solved), it was thought to be essentially solved by the independence results of Goedel and Cohen (and Easton) with some isolated positive results (like Galvin-Hajnal). It was expected that only more independence results remained to be proved. The author has come to change his view: we should stress ]*N0 (not 2] ) and mainly look at the cofinalities rather than cardinalities, in particular pp (), pcf ( ). Their properties are investigated here and conventional cardinal arithmetic is reduced to 2]*N (*N - regular, cases totally independent) and various cofinalities. This enables us to get new results for the conventional cardinal arithmetic, thus supporting the interest in our view. We also find other applications, extend older methods of using normal fiters and prove the existence of Jonsson algebra.

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the fifth publication in the Perspectives in Logic series, studies set-theoretic independence results (independence from the usual set-theoretic ZFC axioms), in particular for problems on the continuum. The author gives a complete presentation of the theory of proper forcing and its relatives, starting from the beginning and avoiding the metamathematical considerations. No prior knowledge of forcing is required. The book will enable a researcher interested in an independence result of the appropriate kind to have much of the work done for them, thereby allowing them to quote general results.

An abstract elementary class (AEe is a class of structures of a fixed vocabulary satisfying some natural closure properties. These classes encompass the normal classes defined in model theory and natural examples arise from mathematical practice, e.g. in algebra not to mention first order and infinitary logics. An AEC is always endowed with a special substructure relation which is not always the obvious one. Abstract elementary classes provide one way out of the cul de sac of the model theory of infinitary languages which arose from over-concentration on syntactic criteria. This is the second volume of a two-volume monograph on abstract elementary classes. It is quite self-contained and deals with three separate issues. The first is the topic of universal classes, i.e. classes of structures of a fixed vocabulary such that a structure belongs to the class if and only if every finitely generated substructure belongs. Then we derive from an assumption on the number of models, the existence of an (almost) good frame. The notion of frame is a natural generalization of the first order concept of superstability to this context. The assumption says that the weak GCH holds for a cardinal $\lambda$, its successor and double successor, and the class is categorical in the first two, and has an intermediate value for the number of models in the third. In particular, we can conclude from this argument the existence of a model in the next cardinal. Lastly we deal with the non-structure part of the topic, that is, getting many non-isomorphic models in the double successor of $ \lambda$ under relevant assumptions, we also deal with almost good frames themselves and some relevant set theory.

An abstract elementary class is a class of structures of the same vocabulary (like a class of rings, or a class of fields), with a partial order that generalizes the relation "A is a substructure (or an elementary substructure) of B." The requirements are that the class is closed under isomorphism, and that isomorphic structures have isomorphic (generalized) substructures; we also require that our classes share some of the most basic properties of elementary classes, like closure under unions of increasing chains of substructures. We would like to classify this general family; in the sense of proving dichotomies: either we can understand the structure of all models in our class or there are many to some extent. More specifically we would like to generalize the theory about categoricity and superstability to this context.