Cardinal Arithmetic [Hardcover]

Is the continuum hypothesis still open? If we interpret it as finding the laws of cardinal arithmetic (or exponentiation, since addition and multiplication were classically solved), the hypothesis would be solved by the independence results of G?del, Cohen, and Easton, with some isolated positive results (like Gavin-Hajnal). Most mathematicians expect that only more independence results remain to be proved. InCardinal Arithmetic, however, Saharon Shelah offers an alternative view. By redefining the hypothesis, he gets new results for the conventional cardinal arithmetic, finds new applications, extends older methods using normal filters, and proves the existence of Jonsson algebra. Researchers in set theory and related areas of mathematical logic will want to read this provocative new approach to an important topic.

1. Basic Confinalities of Small Reduced Products
2. +1 Has a Jonsson Algebra
3. There are Jonsson Algebras in Many Inaccessible Cardinals
4. Jonsson Algebras in Inaccessibles , not -Mahlo
5. Bounding pp( ) when cf( ) using ranks and normal filters
6. Bounds of power of singulars: Induction
7. Strong covering lemma and CH in V[r]
8. Advanced: Cofinalities of reduced products
9. Cardinal Arithmetic
Appendix 1: Colorings
Appendix 2: Entangled orders and narrow Boolean algebras

Saharon Shelah's pcf ( possible cofinalities ) theory and its applications have become a major branch of set-theoretic research since the late 1980's, illuminating many issues involving singular cardinals in combinatorial set theory and the theory of large cardinals. . . .Saharon Shelah is a phenomenal mathematician, preeminent both in model theory and in set theory. His work, beginning in the early 1970's, has advanced both subjects tremendously, and even now, as he enters his fifties, he is going from strength to strength, continuing to produce results at a furious pace. --The Journalof Symbolic Logil