Elements of Mathematics - Theory of Sets

212 Pages·1968·11.941 MB·English

by  Nicolas Bourbaki

ACTUALITll:S SCIENTIFIQUES ET INDUSTRIELLES NICOLAS BOURBAKI
HERMANN I
ADIWES INTERNATIONAL SERIES
IN MATHEMATICS
A. J. Lohwater, Consulting Editor
Elements of Mathematics
Theory of Sets
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HERMANN, PUBLISHERS IN ARTS AND SCIENCE
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115, Boulevard Saint-Germain~ Paris, France
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ADDISON-WESLEY PUBLISHING COMPANY
Reading, Massachusetts-Menlo Park, California-London ·Don Mills, Ontario
A translation of TO THE READER
tl.bmNTS DE MATHEMATIQ.UE, THEORIE DES ENSEMBLES,
originally published in French by Hermann, Paris
I. This series of volumes, a list of which is given on pages VII and vm, 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 modern 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 frequently 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
Library of Congress Catalog card number 68-25177 text, and will prefer not to leave them out, even at a first reading. Their
Printed in France 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
C 1968 by Hermann, all rights reserved.
This book, or parts thereof, may not be reproduced in any form without the publishers"
v
written penn.is:lion.
TO THE READER TO THE READER
assumes as known only those results which have already been discussed The bibliography which follows each historical note contains in general
in the preceding volumes. This rule holds good within each Book, but only those books and original memoirs which have been of the greatest
for convenience of exposition these Books are no longer arranged in a importance in the evolution of the theory under discussion. It makes
consecutive order. At the beginning of each of these Books (or of these no sort of pretence to completeness; in particular, references which serve
chapters), the reader will find a precise indication of its logical relation only to determine questions of priority are almost always omitted.
ship to the other Books and he will thus be able to satisfy himself of the As to the exercices, we have not ·thought it worthwhile in general to
absence of any vicious circle. indicate their origins, since they have been taken from many different
sources (original papers, textbooks, collections of exercises).
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 11. References to a part of this series are given as follows :
mainly to be borne in mind for subsequent use. Less important results a) If reference is made to theorems, axioms, or definitions presented
and those which can easily be deduced from the theorems are labelled as in the same section, they are quoted by their number.
"propositions", "lemmas", "corollaries", "remarks", etc. Those which b) If they occur in another section if the same chapter, this section is also quoted
may be omitted at a first reading are printed in small type. A commentary
in the reference.
on a particularly important theorem appears occasionally under the name
c) If they occur in another chapter in the same Book, the chapter and section
of ''scholium''.
are quoted.
To avoid tedious repetitions it is sometimes convenient to introduce
notations or abbreviations which are in force only within a certain chapter d) If they occur in another Book, this Book is first quoted by its title.
or a certain section of a chapter (for example, in a chapter which is con The Summaries of Results are quoted by the letter R : thus Set Theory,
cerned only with commutative rings, the word "ring" would always R signifies "Summary of Results of the Theory of Sets".
signify "commutative ring"). Such conventions are always explicitly
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
Z
CONTENTS
("dangerous bend").
OF
7. The Exercises are designed both to enable the reader to satisfy himself
THE ELEMENTS OF MATHEMATICS SERIES
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,
I. THEORY OF SETS
except where there appeared to be good reasons for deviating from it.
!. Description of formal mathematics. 2. Theory of sets. 3. Ordered
9. We have made a particular effort always to use rigorously correct
sets; cardinals; natural numbers. 4. Structures.
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. ll. ALGEBRA
I 0. Since in principle the text consists of the dogmatic exposition of a !. Algebraic structures. 2. Linear algebra. 3. Tensor algebras, exterior
theory, it contains in general no references to the literature. Biblio algebras, symmetric algebras. 4. Poylnomials and rational fractions.
graphical references are gathered together in Historical Notes, usually 5. Fields. 6. Ordered groups and fields. 7. Modules over principal ideal
at the end of each chapter. These notes also contain indications, where rings. 8. Semi-simple modules and rings. 9. Sesquilinear and quadratic
appropriate, of the unsolved problems of the theory. forms.
VI VII
•.
' ''
.
CONTENTS OF THE ELEMENTS OF MATHEMATICS SERIES
III. GENERAL TOPOLOGY CONTENTS
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 R•. 8. Complex
numbers. 9. Use of real numbers in general topology. 10. 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.
TO THE READER . , ..... , ........ , , ....... , • , ..... , ... , . . . . . . V
CoNTENTS oF THE ELEMENTS oF MATHEMATics SERIES . . • . . . • . . . . vn
V. TOPOLOGICAL VECTOR SPACES
INTRODT.:CTION • . • • • • . • . • • . • . . • • . • . . . . • . • . • • . • . . • • . . . . • . • . . . • 7
1. Topological vector spaces over a valued field. 2. Convex sets and
locally convex spaces. 3. Spaces of continuous linear mappings. CHAPTER I. DEscRIPTION OF FoRMAL MATHEMATics............. 15
4. Duality in topological vector spaces. 5. Hilbert spaces: elementary
theroy. Dictionary. § I. Terms and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
I. Signs and assemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2. Criteria of substitution... . . . . . . . . . . . . . . . . . . . . . . . . . 18
VI. INTEGRATION 3. Formative constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4. Formative criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1. Convexity inequalities. 2. Riesz spaces. 3. Measures on locally
compact spaces. 4. Extension of a measure. LP spaces. 5. Integration § 2. Theorems ......................................... . 24
of measures. 6. Vectorial integration. 7. Haar measure. 8. Convo-
1. The axioms ................................... . 24
lution and representation.
2. Proofs ........................................ . 25
3. Substitutions in a theory ........................ . 26
Lm GROUPS AND LIE ALGEBRAS 4. Comparison of theories .......................... . 26
I. Lie algebras. § 3. Logical theories .................................... . 28
I. Axioms ....................................... . 28
2. First consequences .............................. . 29
CoMMuTATIVE ALGEBRA 3. Methods of proof .............................. . 30
I. Flat modules. 2. Localization. 3. Graduations, filtrations and topo 4. Conjunction ................................... . 33
logies. 4. Associated prime ideals and primary decomposition. 5. In 5. Equivalence .................................... . 34
tegers. 6. Valuations. 7. Divisors.
§ 4. Quantified theories ................................. . 36
I. Definition of quantifiers .......................... . 36
SPECTRAL THEORIES 2. Axioms of quantified theories .................... . 37
3. Properties of quantifiers .......................... . 38
I. Normed algebras. 2. Locally compact groups. 4. Typical quantifiers ............................. . 41
vm
CONTENTS CONTENTS
§ 5. Equalitarian theories ............................... . 44 8. Retractions and sections 86
I. The axioms ................................... . 44 9. Functions of two argume~~; : : : : : : : : : : : : : : : : : : : : : : 89
45
2. Properties of equality ........................... . § 4. Union and intersection of a family of sets .............. . 90
3. Functional relations ............................ . 47
I. Definition of the union and the intersection of a familv
Appendix. Characterization of terms and relations ......... . 50 of sets ...................................... ' 90
I. Signs and words ................................ . 50 32.. IPmroapgeerst ioesf ao fu unnioionn a anndd a inn tienrtseercsteicotni o~ .............. .. 93
51 94
2. Significant words ............................... . 4. Complements of unions and intersectio~~ ........... .
51 96
3. Characterization of significant words .............. . 5. Union and intersection of two sets .......... .
4. Application to assemblies in a mathematical theory .. 53 96
~: ~~~~~f~!:. :: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 98
Exercises for § I ....................................... . 56 99
8. Sum of a family of sets .......................... . 100
Exercises for § 2 ........................ · ... · · · · . · · · · · · · · 56
§ 5. Product of a family of sets ........................... . 101
57
Exercises for § 3 ..................................... · · ..
I. The axiom of the set of subsets 101
Exercises for § 4 ........................................ . 58 2. Set of mappings of one set into ~~~;he~ : : : : : : : : : : : : : 102
3. Definitions of the product of a family of sets ........ . 103
Exercises for § 5 ........................................ . 59 4. Partial products ................................ . 105
Exercises for the Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5. Associativity of products of sets ................... . 106
6. Distributivity formulae .......................... . 107
CHAPTER II. 'THEORY OF SETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7. Extension of mappings to products ................ . Ill
65 § 6. Equivalence relations 112
§ I. Collectivizing relations ............................. · ·
I. The theory of sets .............................. . 65 I. Definition of an equivalence relation ............... . 113
2. Inclusion ...................................... . 66 2. Equivalence classes; quotient set .................. . 114
3. The axiom of extent ............................ . 67 3. Relations compatible with an equivalence relation ... . 116
4. Collectivizing relations .......................... . 67 4. Saturated subsets ............................... . II 7
5. The axiom of the set of two elements .............. . 69 5. Mappings compatible with equivalence relations .... . 118
6. The scheme of selection and union ............... . 69 6. Inverse image of an equivalence relation· induced
7. Complement of a set. The empty set ............. . 71 equivalence relation .................. : ....... . 119
7. Quotients of equivalence relations ................ . 119
§ 2. Ordered pairs ...................................... . 72 8. Product of two equivalence relations .............. . 120
I. The axiom of the ordered pair ................... . 72 9. Classes of equivalent objects ..................... . 121
2. Product of two sets ............................. . 74
Exercises for § I ........................................ . 123
75
§ 3. Correspondences ............................... · · .. .
Exercises for § 2 ........................................ . 123
I. Graphs and correspondences ..................... . 75
2. Inverse of a correspondence ...................... . 78 Exercises for § 3 ........................................ . 124
78
3. Composition of two correspondences .............. .
4. Functions .................................. · .. . 81 Exercises for § 4 ........................................ . 125
82
5. Restrictions and extensions of functions ........... . Exercises for § 5 ........................................ . 127
6. Definition of a function by means of a term ....... . 83
7. Composition of two functions. Inverse function .... . 84 Exercises for § 6 ........................................ . 127
2 3
CONTENTS CONTENTS
CHAPTER Ill. ORDERED SETS, CARDINALS, INTEGERS •••.•••.•.••• 131 4. Finite sequences ................................ . 175
5. Characteristic functions of sets .................... . 175
§ 1. Order relations. Ordered sets ......................... . 131
6. Euclidean division .............................. . 176
I. Definition of an order relation .................... . 131 7. Expansion to base b ............................. . 177
2. Preorder relations ............................. · . 133 8. Combinatorial analysis .......................... . 179
3. Notation and terminology ....................... . 135
4. Ordered subsets. Product of ordered sets .......... . 136 § 6. Infinite sets ....................................... . 183
5. Increasing mappings ............................ . 138 1. The set of natural integers ....................... . 183
6. Maximal and minimal elements ................... . 139 2. Definition of mappings by induction .............. . 184
7. Greatest element and least element ............... . 140 3. Properties of infinite cardinals .................... . 186
8. Upper and lower bounds ...................... ·. · 141 4. Countable sets .................................. . 188
9. Least upper bound and greatest lower bound ....... . 141 5. Stationary sequences ............................ . 189
I 0. Directed sets ................................. · .. 145
145 § 7. Inverse limits and direct limits ....................... . 191
II. Lattices .............................. · · .. · · · · · ·
12. Totally ordered sets ......................... · · · . · 146 1. Inverse limits .................................. . 191
13. Intervals ...................................... . 147 2. Inverse systems of mappings ..................... . 192
3. Double inverse limit. ............................ . 196
§ 2. Well-ordered sets .................................. . 148 4. C?nditi?n~ for an inverse limit to be non-empty .... . 198
1. Segments of a well-ordered set .................... . 148 5. Direct hm1ts ................................... . 201
2. The principle of transfinite induction .............. . !51 6. Direct systems of mappings ...................... . 204
3. Zermelo's theorem ................... · . · .. · · · · · · · !52 7. Double direct limit. Product of direct limits ....... . 209
4. Inductive sets ............................... · · . · !54
Exercises for § I ........................................ . 212
5. Isomorphisms of well-ordered sets ............. · · · . · !55
6. Lexicographic products ......................... . !57 Exercises for § 2 ........................................ . 221
§ 3. Equipotent sets. Cardinals ................. · · .. · · · · · · · !57 Exercises for § 3 ........................................ . 229
I. The cardinal of a set .......................... .. !57 Exercises for § 4 ........................................ . 230
2. Order relation between cardinals ............... · · . · !58
3. Operations on cardinals ..................... · · · · · 160 Exercises for § 5 ........................................ . 235
4. Properties of the cardinals 0 and I ................ . 162 Exercises for § 6 ........................................ . 241
5. Exponentiation of cardinals ...................... . 163
6. Order relation and operations on cardinals ......... . 164 Exercises for § 7 ........................................ . 251
§ 4. Natural integers. Finite sets .................... · · · · · · · 166 Historical Note on § 5 ................................... . 255
I. Definition of integers ................... · · · · · · · · · · 166 Bibliography ........................................... . 257
2. Inequalities between integers ..................... . 166
3. The principle of induction ....................... . 168 CHAPTER IV. STRUCTURES ...••.••••.•........•............. 259
4. Finite subsets of ordered sets ...................... . 170
5. Properties of finite character ...................... . 170 § 1. Structures and isomorphisms ........................ . 259
1. Echelons ...................................... . 259
§ 5. Properties of integers ................................ . 171 2. Canonical extensions of mappings ................. . 260
I. Operations on integers and finite sets .............. . 171 3. Transportable relations .......................... . 261
2. Strict inequalities between integers ................ . 173 4. Species of structures ............................ . 262
3. Intervals in sets of integers ....................... . 174 5. Isomorphisms and transport of structures .......... . 264
5
4
CONTENTS
INTRODUCTION
266
6. Deduction of structures ............. · . · · · · · · · · · · · ·
7. Equivalent species of structures .......... · · .. · · · · · · 268
271
§ 2. Morphisms and derived structures ... · ... · · · · · · · · · · · · · ·
271
I. Morphisms ............................. · ..... · ·
273
2. Finer structures .............. · · · . · · · · · · · · · · · · · · · ·
274
3. Initial structures .................. · · · · · · · · · · · · · ·
276
4. Examples of initial structures ....... · · · · . · · · · · · · · · ·
280
5. Final structures ............... · .. · · · · · · · · · · · · · ·
281
6. Examples of final structures ........ · · · . · · · · · · · · · ·
283
§ 3. Universal mappings ............. · · ·. · · · · · · · · · · · · · · · ·
283
I. Universal sets and mappings ............ · ... · · · · · ·
284
2. Existence of universal mappings .......... · .. · · · · · · ·
286
3. Examples of universal mappings ... · · · · . · · · · · · · · · · ·
289
Exercises for § I ................... · · · · . · · · · · · · · · · · · · · · Ever since the time of the Greeks, mathematics has involved proof; and
..................... . . . . . . . . . . . . . . . . . . 290 it is even doubted by some whether proof, in the precise and rigorous
Exercises for § 2 sense which the Greeks gave to this word, is to be found outside mathe
Exercises for § 3 ............. · · · · · · · · · · · · · · · · · · · · · · · · · · · 294 matics. Vofe may fairly say that this sense has not changed, because what
constituted a proof for Euclid is still a proof for us; and in times when the
296
Historical Note on Chapters I-IV ..... · . · .. · . · · · · · · · · · · · · · · concept has been in danger of oblivion, and consequently mathematics
343 itself has been threatened, it is to the Greeks that men have turned again
Bibliography ................ · · · · · · · · · · · · · · · · · · · · · · · · · ·
for models of proof. But this venerable bequest has been enlarged during
347 the past hundred years by important acquisitions.
SuMMARY OF RESULTS ................... · · · · · · · · · · · · · · · · · · · · ·
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .................. 347 texBtsy, aint ahlyassi sb eoefn t hpeo mssiebclhea tnoi sdmis ocef rpnr otohfSe isntr usuctituarbel yu ncdheorsleyni nmg abthoethm avtoiccaa l
347
§ 1. Elements and subsets of a set ...... · .. · · · · · · · · · · · · · · · · bulary and syntax. This analysis bas led to the conclusion that a suffi
351
§ 2. Functions ................ · · · · · · · · · · · · · · · · · · · · · · · · ciently explicit mathematical text could be expressed in a conventional
§ 3. Products of sets .................. · · · · · · · · · · · · · · · · · · · 357 language containing only a small number of fixed "words", assembled
§ 4. Union, intersection, product of a family of sets .. · · · · · · · · · 363 according to a syntax consisting of a small number of unbreakable rules :
370 such a text is said to be formalized. The description of a game of chess,
§ 5. Equivalence relations and quotient sets ...... · · · · · · · · · · ·
in the usual notation, and a table oflogarithms, are examples of formalized
§ 6. Ordered sets . . . . . . . . . . . . . . . . . . . . ................ . 373 texts. The formulae of ordinary algebraic calculation would be another
380
§ 7. Powers. Countable sets .................. · · · · · · · · · · example, if the rules governing the use of brackets were to be completely
§ 8. Scales of sets. Structures ............. · · · · · · · · · · · · · · · 383 codified and strictly adhered to; in practice, some of these rules are never
made explicit, and certain derogations of them are allowed.
387
Index of notation ................ · · · · · · · · · · · · · · · · · · · · · · · · The verification of a formalized text is a more or less mechanical process,
Index of terminology ...................... · · · · · · · · · · · · 392 the only possible causes of error being the length or intricacy of the text :
this is why a mathematician will normally accept with confidence the result
Axioms and schemes of the theory of sets ........ · · · · · · · · · · · 414 of an algebraic calculation performed by another, provided he knows
that the calculation is not unduly long and has been done with care.
On the other hand, in an unformalized text, one is exposed to the dangers
of faulty reasoning arising from, for example, incorrect use of intuition
7
6
INTRODUCTION INTRODUCTION
or argument by analogy. In practice, the mathematician who wishes · nother specialization of the general theory). Moreover- and thi~ is
to satisfY himself of the perfect correctness or "rigour" of a proof or a ~h:t concerns us most particularly in this serie~ of volumes - the a~w­
theory hardly ever has recourse to one or another of the complete formali matic method allows us, when we are concerned w1th complex mathematical
zations available nowadays, nor even usually to the incomplete and partial ob · ects, to separate their properties and regroup_ the~ arou.nd a sm~ll
formalizations provided by algebraic and other calculi. In general he m;mber of concepts :that is to say, using a word wh1ch w1ll rece1ve ~ prec1se
is content to bring the exposition to a point where his experience and definition later, to classify them according to the. stru~tures to w~1ch t~ey
mathematical flair tell him that translation into formal language would belong. (Of course, the same structure can anse m connectiOn wJ_th
be no more than an exercise of patience (though doubtless a very tedious · d'fferent mathematical objects.) For example, some of the properties
var1ous I l b · h · be
one). If, as happens again and again, doubts arise as to the correctness of of the sphere are topological, others are age ra1c, ot ers agam can .
the text under consideration, they concern ultimately the possibility of considered as belonging to differential geom~try or the theory of L1e
translating it unambiguously into such a formalized language : either Although this principle may occaswnally become somewhat
groups. . . th b . f
because the same word has been used in different senses according to the artificial when applied to closely interwoven structures, It 1s e as1s o
context, or because the rules of syntax have been violated by the unconscious the division into Books of the subject-matter of th1s senes.
use of modes of argument which they do not specifically authorize, or again
because a material error has been committed. Apart from this last possi ~ Just as the art of speaking a language correctly precedes the invention
bility, the process of rectification, sooner or later, invariably consists in ~f grammar, so the axiomatic method had been pr~ctised long before the
the construction of texts which come closer and closer to a formalized text · vention of formalized languages; but its consciOUS practice can rest
until, in the general opinion of mathematicians, it would be superfluous ~:ly on the knowledge of the general principles ~overning such la'_lguages
to go any further in this direction. In other words, the correctness of a and their relationship with current mathematical t"":ts. In th1~ .Booak:
mathematical text is verified by comparing it, more or less explicitly, our first object is to describe such a language, together w1th a_n exposltlon
with the rules of a formalized language. general principles which could be applied to many ot~er sJmJlar languages,
The axiomatic method is, strictly speaking, nothing but this art of drawing however, one of these languages will always be sufficient for our purpos;s.
up texts whose formalization is straightforward in principle. As such it For whereas in the past it was thought that every branc~ of ~athemat1cs
is not a new invention; but its systematic use as an instrument of discovery depended on its own particular intuitions which. provid;d 1ts conc:pts
is one of the original features of contemporary mathematics. As far as and prime truths, nowadays it is known to be poss~ble, log1ca~ly speaking,
reading or writing a formalized text is concerned, it matters little whether to derive practically the whole of known mathematics from a smgle ~ource,
this or that meaning is attached to the words, or signs in the text, or indeed the Theory of Sets. Thus it is sufficient for our purposes to descnbe the
whether any meaning at all is attached to them; the only important point principles of a single formalized language, to indicate how the Th~ory
is the correct observance of the rules of syntax. Thus, as everyone knows, of Sets could be written in this language, and then to show how the vanous
the same algebraic calculation can be used to solve problems about pounds branches of mathematics, to the extent that we are concerned ;-vith the.m
weight or pounds sterling, about parabolas or motion under gravity. The in this series, fit into this framework. By so doing we do not clmm to .le.gis
same advantage attaches to every text written according to the axiomatic late for all time. It may happen at some future date that mat~ema~Jcians
method, and for the same reasons: once the theorems of general topology will agree to use modes of reasoning which cannot be formahzed m the
have been established, they may be applied at will to ordinary space, language described here : it would then be necessary, if not to change the
Hilbert space, and many others. This faculty of being able to give differ language completely, at least to enlarge its rules of syntax. But that IS
ent meanings to the words or prime concepts of a theory is indeed an for the future to decide. .
important source of enrichment of the mathematician's intuition, which is It goes without saying that the description of the formahzed language
not necessarily spatial or sensory, as is sometimes believed, but is far more is made in ordinary language, just as the rules of chess are. We do .not
a certain feeling for the behaviour of mathematical objects, aided often propose to enter into a discussion of the. psychological a?d metaphysical
by images from very varied sources, but founded above all on everyday problems which underlie the use of ordmary l~~guage m such Circum
experience. Thus one is often led to study with profit those parts of a stances (for example, the possibility of recogmzmg that a letter of the
theory which traditionally have been neglected in this theory but studied alphabet is "the same" in two different places on the page, etc.). Mo_re-
systematically in a general axiomatic context, of which the given theory over, it is scarcely poss1. ble to undertake sue h a d escn· pt·w n W"l th ou tm. akinhg
is a special case (for example, properties which have their historical origin use of numeration. It is objected by some that the use of numbers m sue
9
8
INTRODUCTION INTRODUCTION
a context is suspect, even tantamount to petitio principii. It is clear, however, for example, to the following: "if a bag of counters contains black counters
that in fact we are using numbers merely as marks (and that we could and white counters, and if we replace all the black counters by white
for that matter replace them by other signs, such as colours or letters), ones then there will be only white counters in the bag"). But very
and that we are not making use of any mathematical reasoning when we soon' we meet examples where the argument takes a typically mathe
number the signs which occur in an explicitly written formula. We matical turn, in which the use of arbitrary integers and mathematical
shall not enter into the question of teaching the principles of a formalized induction predominates. Although we have countered the objection made
language to beings whose intellectual development has not reached the earlier against the use of numeration in the description of a formalized
stage of being able to read, write, and count. language, it is no Ionge; possible at this ~oi~t to. deny the danger of begg!ng
the question, because nght from the begmnmg It seems that we are makmg
If formalized mathematics were as simple as the game of chess, then once use of all the resources of arithmetic, of which (among other things) we
our chosen formalized language had been described there would remain propose later to lay down the foundati?ns. To this objec~ion the r~ply
only the task of writing out our proofs in this language, just as the author is sometimes made that arguments of this type merely descnbe operatwns
of a .::hess manual writes down in his notation the games he proposes to which can be actually carried out and verified, and for this reason carry
teach, accompanied by commentaries as necessary. But the matter is conviction of another order from that which can be accorded to mathe
far from being as simple as that, and no great experience is necessary to matics proper. But it seems simpler to say that we could avoid all these
perceive that such a project is absolutely unrealizable : the tiniest proof metamathematical arguments if the formalized text were explicitly written
at the beginning of the Theory of Sets would already require several out : instead of using the "deductive criteria", we should recommence
hundreds of signs for its complete formalization. Hence, from Book I each time the sequences of operations which the purpose of the criteria
of this series onwards, it is imperative to condense the formalized text by is to abbreviate by predicting their result. But formalized mathematics
the introduction of a fairly large number of new words (called abbreviating cannot in practice be written down in full, and therefore we must have
symbols) and additional rules of syntax (called deductive criteria). By doing confidence in what might be called the common sense of the mathemati
this we obtain languages which are much more manageable than the cian : a confidence analogous to that accorded by a calculator or an
formalized language in its strict sense. Any mathematician will agree engineer to a formula or a numerical table without any awareness of the
that these condensed languages can be considered as merely shorthand existence ofPeano's axioms, and which ultimately is based on the knowledge
transcriptions of the original formalized language. But we no longer that it has never been contradicted by facts.
have the certainty that the passage from one of these languages to another We shall therefore very quickly abandon formalized mathematics, but
can be made in a purely mechanical fashion : for to achieve this certainty not before we have carefully traced the path which leads back to it. The
it would be necessary to complicate the rules of syntax which govern the first "abuses of language" thus introduced will allow us to write the rest
use of the new rules to such a point that their usefulness became illusory; of this series (and in particular the Summary of Results of Book I) in
just as in algebraic calculation and in almost all forms of notation commonly the same way as all mathematical texts are written in practice, that is to
used by mathematicians, a workable instrument is preferable to one which say partly in ordinary language and partly in formulae which constitute
is theoretically more perfect but in practice far more cumbersome. partial, particular, and incomplete formalizations, the best-known examples
As the reader will see, the introduction of this condensed language is of which are the formulae of algebraic calculation. Sometimes we shall
accompanied by "arguments" of a particular type, which belong to use ordinary language more loosely, by voluntary abuses of language,
what is called metamathematics. This discipline ignores entirely any by the pure and simple omission of passages which the reader can safely
meaning which may originally have been attributed to the words or phrases be assumed to be able to restore easily for himself, and by indications which
of formalized mathematical texts, and considers these texts as particularly eannot be translated into formalized language and which are designed to
simple objects, namely as assemblies of previously given objects in which help the reader to reconstruct the complete text. Other passages, equally
only the assigned order is of importance. Moreover, just as for example untranslatable into formalized language, are introduced in order to clarify
a chemistry textbook announces in advance the result of an experiment the ideas involved, if necessary by appealing to the reader's intuition;
performed under given conditions, so metamathematical "arguments" this use of the resources of rhetoric is perfectly legitimate, provided only
usually assert that when a succession of operations has been performed on a that the possibility of formalizing the text remains unaltered. The first
text of a given type, then the final text will be of another given type. In examples of this will appear in Chapter III of this Book, which describes
the simplest cases, the assertions are indeed the purest truisms (comparable, the theory of integers and cardinal numbers.
10 II
e•
INTRODUCTION
INTRODUCTION
all rational use of our intellectual faculties. Since the various. mathematical
Thus, written in accordance with the axiomatic method and keeping
theories are now logically attached to the Theory of Sets, it follows that
always in view, as it were on the horizon, the possibility of a complete
any contradiction encountered in one or anothe~ of these theories ~u~t
formalization, our series lays claim to perfect rigour : a claim which is not
·ve rise to a contradiction in the Theory of Sets itself. Of course, thiS iS
in the least contradicted by the preceding considerations, nor by the need ~ot an argument from which we can infer th~ consisten.cy of the. Theory
to correct errors which slip into the text from time to time.
of Sets. Nevertheless, during the half-century smce the axwms of this theory
were first precisely formulated, these axioms have be~n applied to dr.aw
~ We take the same realistic attitude towards the question of consistency
conclusions in the most diverse branches of mathematics Without le~d~ng
{or non-contradiction). This question has been one of the main preoccu
to a contradiction, so that we have grounds for hope that no contradictwn
pations of modern logicians, and is partly responsible f~r the creat~on ?f
will ever arise.
formalized languages (cf. Historical Note). A mathematical theory iS said
If it should turn out otherwise, it would be that the observed contra
to be contradictory if a theorem and its negation have both been proved
diction was inherent in the fundamental principles of the Theory of Sets;
in the theory; from the usual rules of reasoning, which are the basis of the
these principles would therefore require .to be m~dified, if possibl~ without
rules of syntax in formalized languages, it follows then that every theorem
compromising those parts of mathema trcs we WiSh most .to retam, and 1t
is both true and false in the theory, and the theory is consequently of no
is clear that the task of modification would be made eaSier by the use of
interest. If therefore we are involuntarily led to a contradiction, we
the axiomatic method and formalized language, which allow us to formulate
cannot allow it to remain without rendering futile the theory in which it
these principles more precisely and to separate out their cons~quences
occurs.
more clearly. Indeed, this is more or less what has happened m recent
Can we be certain that this will never happen? Vvithout entering into
times when the "paradoxes" of the Theory of Sets were eliminated by
a discussion - which is outside our competence - of the very notion of
adop;ing a formalized language essentially equivalent to that which we
certainty, we may observe that metamathematics can set itself the task
shall describe here· and a similar revision would have to be undertaken
of examining the problems of consistency by its own methods. To say
if this language in' its turn should pro:'e to be. contradicto:y.
that a theory is contradictory means in effect that it contains a correct
formalized proof which leads to the conclusion 0 # 0. Now metama To sum up, we believe that mathematicS lS destmed to survive, and that
the essential parts of this majestic edifice will never collapse as a result
thematics can attempt, using methods of reasoning borrowed from mathe
of the sudden appearance of a contradiction; but we cannot preter;d
matics, to investigate the structure of such a formalized text, in the hope of
that this opinion rests on anything more than expenence. Some will
"proving" that such a text cannot exist. In fact sue~ "proofs" h~ve
say that this is small comfort; but already for two thousand five hundred
been given for certain partial formalized languages, which are less nch
years mathematicians have been correcting their errors to the con_seq~ent
than that which we propose to introduce, but rich enough to express a
enrichment and not impoverishment of their science; and thiS gives
good part of classical mathematics. We may of course reasonably ask
them the right to face the future with serenity.
what in fact has been "proved" in this way : for, if mathematics were
contradictory, then some of its applications to material objects, and in
particular to formalized texts, would stand in danger of being illusory.
To escape this dilemma, the consistency of a formalized language would
have to be "proved" by arguments which could be formalized in a
language less rich and consequently more worthy of confidence; but a
famous theorem of metamathematics, due to Godel, asserts that this is
impossible for a language of the type we shall describe, which is rich
enough in axioms to allow the formulation of the results of classical arith
metic.
On the other hand, in the proofS of "relative" consistency (that. is
to say, those which establish the consistency of a theory on the suppoSitiOn
that some other theory, for example set theory, is non-contradictory),
the metamathematical part of the argument (cf. Chapter I, § 2, no. 4)
is so simple that it seems hardly possible to deny it without renouncing
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