Classic Set Theory: For Guided Independent Study

296 Pages·2017·28.18 MB·English

by  Derek C Goldrei

Classic Set Theory
FOR GUIDED INDEPENDENT STUDY

Classic Set Theory
Derek Goldrei
Open Universitv
CHAPMAN & HALLICRC
A CRC Press Company
Boca Raton London NewYork Washington, D.C.
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Q 1996 by Derek Goldrei
First edition 1996
First CRC Press reprint 1998
Originally published by Chapman & Hall
No claim to original U.S. Government works
International Standard Book Number 0-412-60610-0
Library of Congress Card Number 96-85139
CONTENTS
Preface vii
1 Introduction 1
1.1 Outline of the book 1
1.2 Assumed knowledge 3
2 The Real Numbers 7
2.1 Introduction 7
2.2 Dedekind 's construction 8
2.3 Alternative constructions 17
2.4 The rational numbers 28
3 The Natural Numbers 33
3.1 Introduction 33
3.2 The construction of the natural numbers 38
3.3 Arithmetic 48
3.4 Finite sets 58
4 The Zermelo-Fraenkel Axioms 66
4.1 Introduction 66
4.2 A formal language 71
4.3 Axioms 1 to 3 75
4.4 Axioms 4 to 6 81
4.5 Axioms 7 to 9 92
5 The Axiom of Choice 104
5.1 Introduction 104
5.2 The axiom of choice 107
5.3 The axiom of choice and mathematics 112
5.4 Zorn's lemma 117
6 Cardinals (without the Axiom of Choice) 127
6.1 Introduction 127
6.2 Comparing sizes 128
6.3 Basic properties of ~ and ::5 133
6.4 Infinite sets without AC - countable sets 140
6.5 Uncountable sets and cardinal arithmetic,
without AC 151
7 Ordered Sets 163
7.1 Introduction 163
7.2 Linearly ordered sets 164
7.3 Order arithmetic 177
7.4 Well-ordered sets 188
8 Ordinal Numbers 202
8.1 Introduction 202
8.2 Ordinal numbers 203
8.3 Beginning ordinal arithmetic 218
8.4 Ordinal arithmetic 231
8.5 The Ns 254
9 Set Theory with the Axiom of Choice 263
9.1 Introduction 263
9.2 The well-ordering principle 264
9.3 Cardinal arithmetic and the axiom of choice 267
9.4 The continuum hypothesis 276
Bibliography 281
Index 283
vi
PREFACE
How to use this book
This book is intended to be used by you for independent study, with no other
reading or lectures etc., much along the lines of standard Open University
materials. There are plenty of exercises within the text which we would rec-
ommend you to attempt at that stage of your work. Almost all are intended
to be reasonably straightforward on the basis of what's come before and many
are accompanied by solutions - it's worth reading these solutions as they of-
ten contain further teaching, but do try the exercises first without peeking,
to help you to engage with the material. Those exercises without solutions
might well be very suitable for any tutor to whom you have access to use as
the basis for any continuous assessment of this material, to help you check
that you are making reasonable progress. But beware! Some of the exercises
pose questions for which there is not always a clear-cut answer: these are
intended to provoke debate! In addition there are further exercises located at The book is also peppered with
the end of most sections. These vary from further routine practice to rather notes in the margins, like this!
hard problems: it's well worth reading through these exercises, even if you They consist of comments meant to
be on the fringe of the main text,
don't attempt them, as they often give an idea of some important ideas or
rather than the core of the
results not in the earlier text. Again your tutor, if you have one, can guide
teaching, for instance reminders
you through these. about ideas from earlier in the
book or particularly subjective
If you would like any further reading in textbooks of set theory, there are
opinions of the author.
plenty of good books available which use essentially the same Zermelo-Fraenkel
axiom system, for instance those by Enderton [I], Hamilton [2] and Suppes
[3], all covering roughly the same material as this book, while the books by
Devlin [4] and Moschovakis [5] go some way further, looking at some of the
modem set theory built on the foundations in this book. For a short outline
of many of the key ideas, Halmos [6] is invaluable.
Acknowledgments
I would like to thank all those who have in some way helped me to write this As is customary, I would like to
book. First there are those from whom I initially learnt about set theory when absolve everyone else from any
studying at the University of Oxford: Robin Gandy and Paul Bacsich. Then attaching to anything in this
book or any of its inadequacies.
there are my colleagues at the Open University who have both taught me so
But whether I mean this is entirely
much about maths and the teaching of maths, and encouraged me in the writ-
another matter!
ing of this book, especially Bob Coates. Next there are those who have given
me practical help in its production, not to mention a contract: Nicki Dennis,
Achi Dosanjh and Stephanie Harding at Chapman & Hall, several anonymous
(but not thereby any less deserving of my thanks!) reviewers, and Alison Ca-
dle and Chris Rowley at the Open University, for refining my Lwstyle files.
Richard Leigh copy-edited the book and made many very helpful comments.
Sharon Powell at the Open University helped me greatly with printing the
final version of the book. Last, but foremost in my mind, there are all my
old students at the Open University and at the University of Oxford, espe-
cially the women (and now men!) of Somerville and St. Hugh's Colleges. In
vii
Preface
particular I'd like to thank the following for their comments on parts of the
book: Dorothy Barton, Kirsten Boyd, Cecily Crampin, Adam Durran, Aneirin
Glyn, Alexandra Goulding, Nick Granger, Michael Healey, Sandra Lewis, J-B
Louveaux, David Manlove, Chris Marshall, Nathan Phillips, Alexandra Ralph
and Axel Schairer.
This book is dedicated to all students at both OUs, but especially to
M2090737.
1 INTRODUCTION
1.1 Outline of the book
The language of sets is part of the vocabulary of any student or user of math-
ematics, acquired very early in one's schooling. Words like 'intersection' and
'union7 have become part of everyday mathematics. But the 'set theory' in
this book is about much deeper and more complicated ideas about sets, at the
heart of any discussion of the foundations of mathematics, especially about
the place of 'infinity' in the subject. Many of these deep ideas have their
origin in the work of Georg Cantor, a German mathematician at his most ac- For a very accessible account of
tive in the latter half of the 19th-century. Cantor had many remarkable (and Cantor's life (1845-1918) and work,
controversial) insights to the nature of infinity, producing a theory involving look at the biography by Dauben
two sorts of 'infinite number', each equipped with an arithmetic incorporat- 171.
ing and extending the familiar arithmetic of the natural numbers. Among the In set theory it is customary to use
questions which he was able to pose and resolve were such as: 'natural number' to mean a
non-negative integer.
1. Are there more rational numbers than natural numbers?
2. Are there more irrational real numbers than rational numbers?
3. Are there more points in the real plane than on the real line?
The obvious answer to the first of these questions is probably 'Yes', on the
grounds that the set of natural numbers is a proper subset of the set of rational
numbers - we make no apologies for leaping into use of the everyday language However, if you are unhappy with
of sets! The obvious answer to the third question might similarly seem to be some of the set terminology, don't
'Yes'. In both cases, Cantor's answer was in fact 'No', and one aim of this worry! We shall explain it more
carefully later on.
book is to explain his reasons why.
There might not be such an obvious candidate for the answer to the second Cantor's answer is 'Yes7.
question above it all depends on how much you already know about the real
-
numbers. Another aim of this book is to give an explanation of what the real
numbers are, another aspect of Cantor's work, and part of the work of his
contemporary Richard Dedekind (1831-1916). This explanation leads to the
need to explain even more basic numbers, namely the natural numbers.
Cantor's work on infinity really stemmed from his and others' work on real
analysis, and there are several connections between the issues of infinity and
of the real numbers. First there was the drive in the 19th century to put
the calculus onto a rigorous footing, giving what is now taught as real anal-
ysis: the last stage of this required a firm algebraic description of the reals,
while earlier stages needed to explain away or eliminate uses of infinity, e.g.
00
the infinitely large, like the m in zn, and the infinitely small, namely Infinitesirnuis,i nfinitely small
quantities, played a major part in
n=O
infinitesimals. Secondly the descriptions of the reals provided by Dedekind the theory of calculus, until
and Cantor required the overt handling, as legitimate mathematical objects, banished through the work of
Weierstrass and others in the
of infinite sets. And thirdly mathematical developments in real analysis led
mid-19th century.
to the desire both to study infinite subsets of the real numbers and to carry
Twentieth-century logic, inspired
out infinite processes on such subsets. in part by Cantor's work, has
resurrectid infinitesimals within
The key issue is whether it is legitimate to treat an infinite set as a math-
the context of non-standard
ematical object which one can then use in further constructions. We are SO analysis.