# FOM: Book on Cardinal Arithmetic

Joe Shipman shipman at savera.com
Wed Aug 16 17:09:04 EDT 2000

Thanks to Baldwin and Roslanowski for the references.  The one that is
closest to what I am looking for seems to be the last one.  I went to
Birkhauser's website and found the following information on it (which is
encouraging enough that I'll go to the Fine Hall library tonight and
examine it closely enough to decide whether to buy it)--JS

Introduction to Cardinal Arithmetic
M. Holz, Universität Hannover, Germany
K. Steffens, Universität Hannover, Germany
E. Weitz, Hamburg, Germany
3-7643-6124-7 * 1999 * \$64.50 * Hardcover * 312 pages

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This book is an introduction to modern cardinal arithmetic in the frame
of the axioms of Zermelo-Fraenkel set theory together with the axiom of
choice. A first part describes the classical theory developed by
Bernstein, Cantor, Hausdorff, König and Tarski between 1870 and 1930.
Next, the development in the seventies 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 n the seventies.

This text is the first self-contained introduction to cardinal
arithmetic which also includes pcf theory. It is aimed at undergraduate,
and also postgraduate students as well as researchers who want to
broaden their knowledge of cardinal arithmetic. It gives a relatively
complete survey of results provable in ZFC.

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Contents:

Preface

Introduction

1. Foundations

1.1 The Axioms of ZFC
1.2 Ordinals
1.3 Transfinite Induction and Recursion
1.4 Arithmetic of Ordinals
1.5 Cardinal Numbers and their Elementary Properties
1.6 Infinite Sums and Products
1.7 Further Properties of \kappa\lambda-the Singular Cardinal Hypothesis

1.8 Clubs and Stationary Sets

2. The Galvin-Hajnal Theorem

2.1 Ideals and the Reduction of Relations
2.2 The Galvin-Hajnal Formula
2.3 Applications of the Galvin-Hajnal Formula

3. Ordinal Functions

3.1 Suprema and Cofinalities
3.2 \kappa-rapid Sequences and the Main Lemma of pcf-Theory
3.3 The Definition and Simple Properties of pcf(a)
3.4 The Ideal J<\lambda(a)

4. Approximation Sequences

4.1 The Sets H(T)
4.2 Models and Absoluteness
4.3 Approximation Sequences
4.4 The Skolem Hull in H(T)
4.5 Diamond-Club Sequences

5. Generators of T

5.1 Universal Sequences
5.2 The Existence of Generators
5.3 Properties of Generators

6. The Supremum of pcf m (a)

6.1 Control Functions
6.2 The Supremum of pcfm(a)

7. Local Properties

7.1 The Ideals J*(b) and Jp<\lambda(a)
7.2 Intervals in pcf(a)

8. Applications of pcf-Theory

8.1 Cardinal Estimates
8.2 Jónsson Algebras
8.3 A Partition Theorem of Todorcevic
8.4 Cofinalities of Partial Orderings ([\lambda] < or =\kappa,\subseteq)

9. The Cardinal Function pp(l)

9.1 pp(\lambda) and the Theorems of Galvin, Hajnal and Silver
9.2 The Lemma of Galvin and Hajnal for pp(\lambda)

Literature

List of Symbols

Index