Introduction to Cardinal Arithmetic

313 Pages·1999·1.65 MB·English

by  M. Holz

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M. Holz
K. Steffens
E. Weitz
Introduction to
Cardinal Arithmetic
Reprint of the 1999 Edition
Birkhäuser Verlag
· ·
Basel Boston Berlin
Authors:
Michael Holz Edmund Weitz
Unter den Bäumchen 17 Bernadottestr. 38
30926 Seelze 22763 Hamburg
Germany Germany
e-mail: [email protected]
Karsten Steffens
Institut für Algebra, Zahlentheorie und
Diskrete Mathematik
Universität Hannover
Welfengarten 1
30167 Hannover
Germany
e-mail: [email protected]
Originally published under the same title in the Birkhäuser Advanced Texts –
Basler Lehrbücher series by Birkhäuser Verlag, Switzerland, ISBN 978-3-7643-6124-7
© 1999 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland
1991 Mathematics Subject Classification 04-01, 04A10, 03E10
Library of Congress Control Number: 2009937809
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Foundations
1.1 The Axioms of ZFC . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Transfinite Induction and Recursion . . . . . . . . . . . . . . . 20
1.4 Arithmetic of Ordinals . . . . . . . . . . . . . . . . . . . . . . . 30
1.5 Cardinal Numbers and their Elementary Properties . . . . . . . 40
1.6 Infinite Sums and Products . . . . . . . . . . . . . . . . . . . . 58
1.7 Further Properties of κλ – the Singular Cardinal Hypothesis . . 70
1.8 Clubs and Stationary Sets . . . . . . . . . . . . . . . . . . . . . 79
1.9 The Erdo¨s-Rado Partition Theorem . . . . . . . . . . . . . . . 96
2 The Galvin-Hajnal Theorem
2.1 Ideals and the Reduction of Relations . . . . . . . . . . . . . . 103
2.2 The Galvin-Hajnal Formula . . . . . . . . . . . . . . . . . . . . 108
2.3 Applications of the Galvin-Hajnal Formula . . . . . . . . . . . 121
3 Ordinal Functions
3.1 Suprema and Cofinalities . . . . . . . . . . . . . . . . . . . . . 131
3.2 κ-rapid Sequences and the Main Lemma of pcf-Theory . . . . . 143
3.3 The Definition and Simple Properties of pcf(a) . . . . . . . . . 151
3.4 The Ideal J (a) . . . . . . . . . . . . . . . . . . . . . . . . . . 159
<λ
4 Approximation Sequences
4.1 The Sets H(Θ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.2 Models and Absoluteness . . . . . . . . . . . . . . . . . . . . . 176
4.3 Approximation Sequences . . . . . . . . . . . . . . . . . . . . . 196
4.4 The Skolem Hull in H(Θ) . . . . . . . . . . . . . . . . . . . . . 199
4.5 (cid:2)-Club Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 202
vi Contents
5 Generators of J (a)
<λ+
5.1 Universal Sequences . . . . . . . . . . . . . . . . . . . . . . . . 209
5.2 The Existence of Generators . . . . . . . . . . . . . . . . . . . . 213
5.3 Properties of Generators . . . . . . . . . . . . . . . . . . . . . . 218
6 The Supremum of pcf (a)
μ
6.1 Control Functions . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.2 The Supremum of pcf (a) . . . . . . . . . . . . . . . . . . . . . 225
μ
7 Local Properties
7.1 The Ideals J (b) and Jp (a) . . . . . . . . . . . . . . . . . . . 233
∗ <λ
7.2 Intervals in pcf(a) . . . . . . . . . . . . . . . . . . . . . . . . . 244
8 Applications of pcf-Theory
8.1 Cardinal Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.2 Jo´nsson Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.3 A Partition Theorem of Todor˘cevi´c . . . . . . . . . . . . . . . . 262
8.4 Cofinalities of Partial Orderings ([λ]≤κ,⊆) . . . . . . . . . . . . 265
9 The Cardinal Function pp(λ)
9.1 pp(λ) and the Theorems of Galvin, Hajnal and Silver . . . . . 270
9.2 The Lemma of Galvin and Hajnal for pp(λ) . . . . . . . . . . . 275
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
Preface
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,
describestheclassical cardinal arithmeticdueto Bernstein,Cantor, Hausdorff,
Ko¨nig,and Tarski. Theresults werefound in theyears between1870 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.
The enthusiasm of our students for cardinal arithmetic contributed much
to the formation of this book. Our thanks are due to S. Shelah for answering
several questions and to T. Jech for sending us some papers. Our thanks are
also due to S. Neumann for many hints to Chapter 9 and to T. Espley and
A. Truong for their contributions concerning the readability of this book.

Introduction
If M and N are sets1 and if there exists a bijection2 from M onto N, then
we say that M and N are equinumerous, and write M ≈ N. To measure the
number of members of a set, we will introduce sets of comparison. With ω we
denote the set of natural numbers; 0 is a natural number. If for example M is
equinumerous to the set {n∈ω : n< 25}, then we say that M has exactly 25
members, and {n∈ω :n<25} is a set of comparison for M. If N is a set and
if N and ω are equinumerous, then ω will be a set of comparison for N, and
N will be called countably infinite or denumerable. A well known example for
such a set is N ={n∈ω :n is divisible by 2}.
We will assign to each set M a set of comparison |M|, the so-called car-
dinality of M, such that the following holds.
(1) M ≈N ⇐⇒|M|=|N|.
(2) M ≈|M|.
Chapter 1, § 1.5 shows that such an assignment M (cid:7)→ |M| is possible. Any
set |M| is called a cardinal number or a cardinal. If |M| and |N| are cardinal
numbers,thenwewrite|M|≤|N|iff3 thereisaninjection4 fromM intoN,and
|M|<|N|iff|M|≤|N|and¬(M ≈N).Ifweassumetheaxiomofchoice,then
any two cardinal numbers are comparable, i.e., we get |M|≤|N| or |N|≤|M|
for any sets M and N; conversely it will not prove difficult to show that this
assertion implies the axiom of choice. Thus the class CN of cardinal numbers
is linearly ordered by ≤ and by <. These linear orderings have no greatest
element. G. Cantor proved in 1873 that, for any set M, the so-called power set
P(M) of M, i.e., the set {x : x ⊆ M}, satisfies |M| < |P(M)|. In Section 1.5
1We assume that the reader has an intuitive knowledge about the mathematical notion
ofaset.{x:E(x)}willbe,asusual,thecollectionofallmathematicalobjectssatisfyingthe
property E(x).
2That is a function f from M into N such that, for every y ∈ N, there is exactly one
x∈M withy=f(x).
3Weusethemathematicalword iffasusual asanabbreviationfor if and only if.
4That is a function f from M into N such that, for every y ∈ N, there is at most one
x∈M withy=f(x).