Saharon Shelah - Böcker

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 Gödel 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 Π]*N0 (not 2]Π) and mainly look at the cofinalities rather than cardinalities, in particular pp (μ), pcf (α). 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.