Ebook: Introduction to Cardinal Arithmetic

This book is an introduction to modern cardinal arithmetic, developed in the frame of the axioms of Zermelo-Fraenkel set theory together with the axiom of choice. It splits into three parts. Part one, which is contained in Chapter 1, describes the classical cardinal arithmetic due to Bernstein, Cantor, Hausdorff, Konig, and Tarski. The results were found in the years between 1870 and 1930. Part two, which is Chapter 2, characterizes the development of cardinal arith­ metic in the seventies, which was led by Galvin, Hajnal, and Silver. The third part, contained in Chapters 3 to 9, presents the fundamental investigations in pcf-theory which has been developed by S. Shelah to answer the questions left open in the seventies. All theorems presented in Chapter 3 and Chapters 5 to 9 are due to Shelah, unless otherwise stated. We are greatly indebted to all those set theorists whose work we have tried to expound. Concerning the literature we owe very much to S. Shelah's book [Sh5] and to the article by M. R. Burke and M. Magidor [BM] which also initiated our students' interest for Shelah's pcf-theory.

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This book is an introduction into modern cardinal arithmetic in the frame of the axioms of Zermelo-Fraenkel set theory together with the axiom of choice (ZFC). A first part describes the classical theory developed by Bernstein, Cantor, Hausdorff, K?nig and Tarski between 1870 and 1930. Next, the development in the 1970s led by Galvin, Hajnal and Silver is characterized. The third part presents the fundamental investigations in pcf theory which have been worked out by Shelah to answer the questions left open in the 1970s.

Reviews:

"The authors aim their text at beginners in set theory. They start literally from the axioms and prove everything they need. The result is an extremely useful text and reference book which is also very pleasant to read." - The Bulletin of Symbolic Logic

"The book should be required reading for every advanced graduate student of set theory. Several courses at various levels could be based on the earlier chapters. There is a useful set of exercises at the end of most sections in the first four chapters." - Mathematical Reviews

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This book is an introduction into modern cardinal arithmetic in the frame of the axioms of Zermelo-Fraenkel set theory together with the axiom of choice (ZFC). A first part describes the classical theory developed by Bernstein, Cantor, Hausdorff, K?nig and Tarski between 1870 and 1930. Next, the development in the 1970s led by Galvin, Hajnal and Silver is characterized. The third part presents the fundamental investigations in pcf theory which have been worked out by Shelah to answer the questions left open in the 1970s.

Reviews:

"The authors aim their text at beginners in set theory. They start literally from the axioms and prove everything they need. The result is an extremely useful text and reference book which is also very pleasant to read." - The Bulletin of Symbolic Logic

"The book should be required reading for every advanced graduate student of set theory. Several courses at various levels could be based on the earlier chapters. There is a useful set of exercises at the end of most sections in the first four chapters." - Mathematical Reviews

Content:
Front Matter....Pages i-vii
Introduction....Pages 1-4
Foundations....Pages 5-102
The Galvin-Hajnal Theorem....Pages 103-127
Ordinal Functions....Pages 129-171
Approximation Sequences....Pages 173-207
Generators of J <?+(a)....Pages 209-220
The Supremum of pcf?(a)....Pages 221-232
Local Properties....Pages 233-248
Applications of pcf-Theory....Pages 249-268
The Cardinal Function pp(?)....Pages 269-286
Back Matter....Pages 287-304

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This book is an introduction into modern cardinal arithmetic in the frame of the axioms of Zermelo-Fraenkel set theory together with the axiom of choice (ZFC). A first part describes the classical theory developed by Bernstein, Cantor, Hausdorff, K?nig and Tarski between 1870 and 1930. Next, the development in the 1970s led by Galvin, Hajnal and Silver is characterized. The third part presents the fundamental investigations in pcf theory which have been worked out by Shelah to answer the questions left open in the 1970s.

Reviews:

"The authors aim their text at beginners in set theory. They start literally from the axioms and prove everything they need. The result is an extremely useful text and reference book which is also very pleasant to read." - The Bulletin of Symbolic Logic

"The book should be required reading for every advanced graduate student of set theory. Several courses at various levels could be based on the earlier chapters. There is a useful set of exercises at the end of most sections in the first four chapters." - Mathematical Reviews

Content:
Front Matter....Pages i-vii
Introduction....Pages 1-4
Foundations....Pages 5-102
The Galvin-Hajnal Theorem....Pages 103-127
Ordinal Functions....Pages 129-171
Approximation Sequences....Pages 173-207
Generators of J <?+(a)....Pages 209-220
The Supremum of pcf?(a)....Pages 221-232
Local Properties....Pages 233-248
Applications of pcf-Theory....Pages 249-268
The Cardinal Function pp(?)....Pages 269-286
Back Matter....Pages 287-304
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