Cardinal Arithmetic

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 P]N0 (not 2]P) and mainly look at the cofinalities rather than cardinalities, in particular pp (µ), pcf (a). Their properties are investigated hereand 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.

“This is a very important book. It is essential reading for anyone working in set theory and its applications.”

The mathematics here will remain an important summit of the subject and the Editors have the good fortune of having obtained a landmark volume for the Logic Guide Series. Proceedings of the Edinburgh Mathematical Society 1998 (41) This book is a great step forward in the development of set theory. Mathematical Reviews Clipping. Bull.London Math.Soc.

Saharon Shelah is at The Hebrew University of Jerusalem.

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 P]N0 (not 2]P) and mainly look at the cofinalities rather than cardinalities, in particular pp (µ), pcf (a). 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.

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