Introduction to Cardinal Arithmetic

by Weitz, E.; Steffens, Karsten; Holz, Michael

• ISBN13: 9783764361242
• ISBN10: 3764361247
• Edition: 1st
• Format: Hardcover
• Copyright: 1999-08-01
• Publisher: Springer Verlag

Summary

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. A first part describes the classical theory developed by Bernstein, Cantor, Hausdorff, Kvnig 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 in the seventies. This text is the first self-contained introduction to cardinal arithmetic which also includes pcf theory. It is aimed at undergraduates, and also at postgraduate students and researchers who want to broaden their knowledge of cardinal arithmetic. It gives a relatively complete survey of results provable in ZFC.

Table of Contents

Preface	vii
Introduction	1	(4)
Foundations
The Axionms of ZFC	5	(10)
Ordinals	15	(5)
Transfinite Induction and Recursion	20	(10)
Arithmetic of Ordinals	30	(10)
Cardinal Numbers and their Elementary Properties	40	(18)
Infinite Sums and Products	58	(12)
Further Properties of κλ the Singular Cardinal Hypothesis	70	(9)
Clubs and Stationary Sets	79	(17)
The Erdos-Rado Partition Theorem	96	(7)
The Galvin-Hajnal Theorem
Ideals and the Reduction of Relations	103	(5)
The Galvin-Hajnal Formula	108	(13)
Applications of the Galvin-Hajnal Formula	121	(10)
Ordinal Functions
Suprema and Cofinalities	131	(12)
κ-rapid Sequences and the Main Lemma of pef-Theory	143	(8)
The Definition and Simple Properties of pef(a)	151	(8)
The Ideal J<λ (a)	159	(14)
Approximation Sequences
The Sets H (Θ)	173	(3)
Models and Absoluteness	176	(20)
Approximation Sequences	196	(3)
The Skolem Hull in H (Θ)	199	(3)
k-Club Sequences	202	(7)
Generators of J<λ+ (a)
Universal Sequences	209	(4)
The Existence of Generators	213	(5)
Properties of Generators	218	(3)
The Supremum of pcfμ (a)
Control Functions	221	(4)
The Supremum of pcfμ (a)	225	(8)
Local Properties
The Ideals J* (b) and Jp<λ (a)	233	(11)
Intervals in pcf (a)	244	(5)
Applications of pcf-Theory
Cardinal Estimates	249	(2)
Jonsson Algebras	251	(11)
A Partition Theorem of Todorcevic	262	(3)
Cofinalities of Partial Orderings ([λ]≤κ,⊆)	265	(5)
The Cardinal Function pp (λ)
pp(λ) and the Theorems of Galvin, Hajnal and Silver	270	(5)
The Lemma of Galvin and Hajnal for pp (λ)	275	(12)
Bibliography	287	(2)
List of Symbols	289	(4)
Index	293