Elements of Mathematics. Theory of Sets

418 Pages·2004·35.26 MB·English

by  Bourbaki, Nicolas

ELEMENTS OF MATHEMATICS
NICOLAS BOURBAKI
ELEMENTS OF MATHEMATICS
Theory of Sets
~ Springer
Originally published as
mements de Mathematique.
Theorie des Ensembles
Paris, 1970
eN.Bourbaki
Mathematics Subject Classification (2000): 03-02, 03Bxx
Library of Congress Control Number: 2004110815
Softcover edition of the 1st printing 1968
ISBN-13:978-3-540-22525-6 e-ISBN-13: 978-3-642-59309-3
DOl: 10.1007/978-3-642-59309-3
Springer Berlin Heidelberg New York
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TO THE READER
I. This series of volumes, a list of which is given on pages vn and VIU, takes
up mathematics at the beginning, and gives complete proofs. In prin
ciple, it requires no particular knowledge of mathematics on the reader's
part, but only a certain familiarity with mathematical reasoning and a
certain capacity for abstract thought. Nevertheless, it is directed especially
to those who have a good knowledge of at least the content of the first
year or two of a university mathematics course.
2. The method of exposition we have chosen is axiomatic and abstract,
and normally proceeds from the general to the particular. This choice
has been dictated by the main purpose of the treatise, which is to provide
a solid foundation for the whole body of modem mathematics. For this
it is indispensable to become familiar with a rather large number of very
general ideas and principles. Moreover, the demands of proof impose
a rigorously fixed order on the subject matter. It follows that the utility
of certain considerations will not be immediately apparent to the reader
unless he has already a fairly extended knowledge of mathematics; other
wise he must have the patience to suspend judgment until the occasion
arises.
3. In order to mitigate this disadvantage we have frequendy inserted
examples in the text which refer to facts the reader may already know
but which have not yet been discussed in the series. Such examples
* ... *.
are always placed between two asterisks: Most readers will
undoubtedly find that these examples will help them to understand the
text, and will prefer not to leave them out, even at a first reading. Their
omission would of course have no disadvantage, from a purely logical
point of view.
4. This series is divided into volumes (here called " Books "). The first
six Books are numbered and, in general, every statement in the text
v
TO THE READER
assumes as known only those results which have already been discussed
in the preceding volumes. This rule holds good within each Book, but
for convenience of exposition these Books are no longer arranged in a
consecutive order. At the beginning of each of these Books (or of these
chapters), the reader will find a precise indication of its logical relation
ship to the other Books and he will thus be able to satisfy himself of the
absence of any vicious circle.
5. The logical framework of each chapter consists of the definitions, the
axioms, and the theorems of the chapter. These are the parts that have
mainly to be borne in mind for subsequent use. Less important results
and those which can easily be deduced from the theorems are labelled as
"propositions", "lemmas", "corollaries", "remarks", etc. Those which
may be omitted at a first reading are printed in small type. A commentary
on a particularly important theorem appears occasionally under the name
of "scholium".
To avoid tedious repetitions it is sometimes convenient to introduce
notations or abbreviations which are in force only within a certain chapter
or a certain section of a chapter (for example, in a chapter which is con
cerned only with commutative rings, the word "ring" would always
signify "commutative ring"). Such conventions are always explicidy
mentioned, generally at the beginning of the chapter in which they occur.
6. Some passages in the text are designed to forewarn the reader against
serious errors. These passages are signposted in the margin with the sign
2
("dangerous bend").
7. The Exercises are designed both to enable the reader to satisfY himself
that he has digested the text and to bring to his notice results which have
no place in the text but which are nonetheless of interest. The most
difficult exercises bear the sign ~.
8. In general, we have adhered to the commonly accepted terminology,
except where there appeared to be good reasons for deviating from it.
9. We have made a particular effort always to use rigorously correct
language, without sacrificing simplicity. As far as possible we have
drawn attention in the text to abuses of language, without which any mathe
matical text runs the risk of pedantry, not to say unreadability.
10. Since in principle the text consists of the dogmatic exposition of a
theory, it contains in general no references to the literature. Biblio
graphical references are gathered together in Historical Notes, usually
at the end of each chapter. These notes also contain indications, where
appropriate, of the unsolved problems of the theory.
VI
TO THE READER
The bibliography which follows each historical note contains in general
only those books and original memoirs which have been of the greatest
importance in the evolution of the theory under discussion. It makes
no sort of pretence to completeness; in particular, references which serve
only to determine questions of priority are almost always omitted.
As to the exercices, we have not thought it worthwhile in general to
indicate their origins, since they have been taken from many different
sources (original papers, textbooks, collections of exercises).
11. References to a part of this series are given as follows :
a) If reference is made to theorems, axioms, or definitions presented
in the same section, they are quoted by their number.
b) If they occur in another section oj the same chapter, this section is also quoted
in the reference.
c) If they occur in another chapter in the same Book, the chapter and section
are quoted.
d) If they occur in another Book, this Book is first quoted by its tide.
The Summaries of Results are quoted by the letter R : thus Set Theory,
R signifies "Summary oj Results oj the Theory oj Sets".
CONTENTS
OF
THE ELEMENTS OF MATHEMATICS SERIES
I. THEORY OF SETS
1. Description of formal mathematics. 2. Theory of sets. 3. Ordered
sets; cardinals; natural numbers. 4. Structures.
II. ALGEBRA
1. Algebraic structures. 2. Linear algebra. 3. Tensor algebras, exterior
algebras, symmetric algebras. 4. Poylnomia1s and rational fractions.
5. Fields. 6. Ordered groups and fields. 7. Modules over principal ideal
rings. 8. Semi-simple modules and rings. 9. Sesquilinear and quadratic
forms.
vn
CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES
III. GENERAL TOPOLOGY
1. Topological structures. 2. Uniform structures. 3. Topological groups.
4. Real numbers. 5. One-parameter groups. 6. Real number spaces,
affine and projective spaces. 7. The additive groups Rli. 8. Complex
numbers. 9. Use of real numbers in general topology. lO. Function
spaces.
IV. FUNCTIONS OF A REAL VARIABLE
1. Derivatives. 2. Primitives and integrals. 3. Elementary functions.
4. Differential equations. 5. Local study of functions. 6. Generalized
Taylor expansions. The Euler-Maclaurin summation formula. 7. The
gamma function. Dictonary.
V. TOPOLOGICAL VECTOR SPACES
1. Topological vector spaces over a valued field. 2. Convex sets and
locally convex spaces. 3. Spaces of continuous linear mappings.
4. Duality in topological vector spaces. 5. Hilbert spaces: elementary
theroy. Dictionary.
VI. INTEGRATION
1. Convexity inequalities. 2. Riesz spaces. 3. Measures on locally
compact spaces. 4. Extension of a measure. LP spaces. 5. Integration
of measures. 6. Vectorial integration. 7. Haar measure. 8. Convo
lution and representation.
Lm GROUPS AND Lm ALGEBRAS
1. Lie algebras.
COMMUTATIVE ALGEBRA
I. Flat modules. 2. Localization. 3. Graduations, filtrations and topo
logies. 4. Associated prime ideals and primary decomposition. 5. In
tegers. 6. Valuations. 7. Divisors.
SPECTRAL THEORmS
1. Normed algebras. 2. Locally compact groups.
vm
CONTENTS
TO THE READER ••..••.....•..•..•.••..••••...•.••••.••..••• V
CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES . . . . • • • . . . . VII
INTRODUCTION ...••.••.••.•..••..•.•.•••••••••.••.•.••••.••. 7
CHAPTER 1. DESCRIPTION OF FORMAL MATHEMATICS... ••••••••.• 15
§ 1. Terms and relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1. Signs and assemblies ............................. 15
2. Criteria of substitution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3. Formative constructions. . . . . . . . . . . . . . . . . . . . . . . . . . 19
4. Formative criteria ............................... 21
§ 2. Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1. The axioms .................................... 24
2. Proofs ......................................... 25
3. Substitutions in a theory ......................... 26
4. Comparison of theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
§ 3. Logical theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1. Axioms ........................................ 28
2. First consequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3. Met?ods .of proof. ............. .... ... ..... ... .. 30
4. ConjunctIon .................................... 33
5. Equivalence..................................... 34
§ 4. Quantified theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1. Definition of quantifiers. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2. Axioms of quantified theories ..................... 37
3. Properties of quantifiers. . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4. Typical quantifiers .............................. 41
CONTENTS
§ 5. Equalitarian theories ................................ 44
1. The axioms .................................... 44
2. Properties of equality ............................ 45
3. Functional relations ............................. 47
Appendix. Characterization of terms and relations . . . . . . . . . . 50
1. Signs and words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2. Significant words.. .... . ............ .... .... ..... 51
3. Characterization of significant words ............... 51
4. Application to assemblies in a mathematical theory .. 53
Exercises for § 1 ........................................ 56
Exercises for § 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Exercises for § 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Exercises for § 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Exercises for § 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Exercises for the Appendix 60
CHAPTER II. THEORY OF SETS 65
§ 1. Collectivizing relations .. .. . . .. . .. .. . . . . . .. .. . .. .. . . . . 65
1. The theory of sets ............................... 65
2. Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3. The axiom of extent ............................. 67
4. Collectivizing relations ........................... 67
5. The axiom of the set of two elements. . . . . . . . . . . . . . . 69
6. The scheme of selection and union ................ 69
7. Complement ofa set. The empty set.............. 71
§ 2. Ordered pairs...... . .. . ... . ........ .... .... .... . .. .. 72
1. The axiom of the ordered pair .................... 72
2. Product of two sets .............................. 74
§ 3. Correspondences.................................... 75
1. Graphs and correspondences ...................... 75
2. Inverse of a correspondence . . . . . . . . . . . . . . . . . . . . . . . 78
3. Composition of two correspondences ............... 78
4. Functions ...................................... 81
5. Restrictions and extensions of functions . . . . . . . . . . . . 82
6. Definition of a function by means of a term. . . . . . . . 83
7. Composition of two functions. Inverse function ..... 84
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