An introduction to the theory of functional equations and inequalities

Marek Kuczma
An Introduction to the
Theory of Functional Equations
and Inequalities
Cauchy’s Equation and Jensen’s Inequality
Second Edition
Edited by
Attila Gilányi
Birkhäuser
Basel · Boston · Berlin
2000 Mathematical Subject Classification: 39B05, 39B22, 39B32, 39B52, 39B62, 39B82,
26A51, 26B25
The first edition was published in 1985 by Uniwersytet Slaski (Katowicach) (Silesian
University of Katowice) and Pánstwowe Wydawnictwo Naukowe (Polish Scientific Publishers)
© Uniwersytet Slaski and Pánstwowe Wydawnictwo Naukowe
Library of Congress Control Number: 2008939524
Bibliographic information published by Die Deutsche Bibliothek
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detailed bibliographic data is available in the Internet at <>.
ISBN 978-3-7643-8748-8 Birkhäuser Verlag AG, Basel – Boston – Berlin
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Editor:
Attila Gilányi
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University of Debrecen
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´
´
Preface to the Second Edition
The first edition of Marek Kuczma’s book An Introduction to the Theory of Func-
tional Equations and Inequalities was published more than 20 years ago. Since then
it has been considered as one of the most important monographs on functional equa-
tions, inequalities and related topics. As J´anos Acz´el wrote in Mathematical Reviews
“ this is a very useful book and a primary reference not only for those working in
functional equations, but mainly for those in other fields of mathematics and its appli-
cations who look for a result on the Cauchy equation and/or the Jensen inequality.”
Based on the considerably high demand for the book, which has even increased
after the first edition was sold out several years ago, we have decided to prepare its
second edition. It corresponds to the first one and keeps its structure and organization
almost everywhere. The few changes which were made are always marked by footnotes.
Several colleagues helped us in the preparation of the second edition. We cor-
dially thank Roman Ger for his advice and help during the whole publication process,
Karol Baron and Zolt´an Boros for their conscientious proofreading, and Szabolcs
Baj´ak for typing and continuously correcting the manuscript. We are grateful to
Eszter Gselmann, Fruzsina M´esz´aros, Gy¨ongyv´er P´eter and P´al Burai for typesetting

several chapters, and we would like to thank the publisher, Birkh¨auser, for undertak-
ing and helping with the publication.
The new edition of Marek Kuczma’s book is paying tribute to the memory
of the highly respected teacher, the excellent mathematician and one of the most
outstanding researchers of functional equations and inequalities.
Debrecen, October 2008
Attila Gil´anyi
Contents
Introduction xiii
Part I Preliminaries
1 Set Theory
1.1 AxiomsofSetTheory 3
1.2 Orderedsets 5
1.3 Ordinalnumbers 6
1.4 Setsofordinalnumbers 8
1.5 Cardinalityofordinalnumbers 10
1.6 Transfinite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 TheZermelotheorem 14
1.8 LemmaofKuratowski-Zorn 15
2 Topology
2.1 Category 19
2.2 Baireproperty 23
2.3 Borelsets 25
2.4 The space z 28
2.5 Analyticsets 32
2.6 OperationA 35
2.7 Theorem of Marczewski . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.8 Cantor-Bendixsontheorem 39
2.9 TheoremofS.Piccard 42

3 Measure Theory
3.1 Outerandinnermeasure 47
3.2 Lineartransforms 54
3.3 Saturatednon-measurablesets 56
3.4 Lusinsets 59
3.5 Outerdensity 61
3.6 Somelemmas 63
viii Contents
3.7 TheoremofSteinhaus 67
3.8 Non-measurablesets 71
4Algebra
4.1 Linear independence and dependence . . . . . . . . . . . . . . . . . . . 75
4.2 Bases 78
4.3 Homomorphisms 83
4.4 Cones 87
4.5 Groupsandsemigroups 89
4.6 Partitionsofgroups 95
4.7 Ringsandfields 98
4.8 Algebraic independence and dependence . . . . . . . . . . . . . . . . . 101
4.9 Algebraicandtranscendentalelements 103
4.10Algebraicbases 105
4.11Simpleextensionsoffields 106
4.12Isomorphismoffieldsandrings 108
Part II Cauchy’s Functional Equation and Jensen’s Inequality
5 Additive Functions and Convex Functions
5.1 Convexsets 117
5.2 Additivefunctions 128
5.3 Convexfunctions 130
5.4 Homogeneityfields 137
5.5 Additive functions on product spaces . . . . . . . . . . . . . . . . . . . 138

5.6 Additive functions on C 139
6 Elementary Properties of Convex Functions
6.1 Convexfunctionsonrationallines 143
6.2 Local boundedness of convex functions . . . . . . . . . . . . . . . . . . 148
6.3 Thelowerhullofaconvexfunctions 150
6.4 TheoremofBernstein-Doetsch 155
7 Continuous Convex Functions
7.1 Thebasictheorem 161
7.2 Compositionsandinverses 162
7.3 Differencesquotients 164
7.4 Differentiation 168
7.5 Differentialconditionsofconvexity 171
7.6 Functionsofseveralvariables 174
7.7 Derivativesofafunction 177
7.8 Derivativesofconvexfunctions 180
7.9 Differentiability of convex functions . . . . . . . . . . . . . . . . . . . . 188
7.10Sequencesofconvexfunctions 192
Contents ix
8 Inequalities
8.1 Jenseninequality 197
8.2 Jensen-Steffenseninequalities 201
8.3 Inequalitiesformeans 208
8.4 Hardy-Littlewood-P´olyamajorizationprinciple 211
8.5 Lim’sinequality 214
8.6 Hadamardinequality 215
8.7 Petrovi´cinequality 217
8.8 Mulholland’sinequality 218
8.9 Thegeneralinequalityofconvexity 223
9 Boundedness and Continuity of Convex Functions and Additive Functions
9.1 The classes A,B,C 227

9.2 Conservativeoperations 229
9.3 Simpleconditions 231
9.4 Measurability of convex functions . . . . . . . . . . . . . . . . . . . . . 241
9.5 Planecurves 242
9.6 Skewcurves 244
9.7 Boundedness below . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
9.8 Restrictions of convex functions and additive functions . . . . . . . . . 251
10 The Classes A, B, C
10.1AHahn-Banachtheorem 257
10.2 The class B 260
10.3 The class C 266
10.4 The class A 267
10.5Set-theoreticoperations 269
10.6 The classes D 271
10.7 The classes A
C
and B
C
276
11 Properties of Hamel Bases
11.1Generalproperties 281
11.2Measure 282
11.3Topologicalproperties 285
11.4Burstinbases 285
11.5 Erd˝ossets 288
11.6Lusinsets 294
11.7Perfectsets 299
11.8 The operations R and U 301
x Contents
12 Further Properties of Additive Functions and Convex Functions

12.1Graphs 305
12.2Additivefunctions 308
12.3Convexfunctions 313
12.4Biggraph 316
12.5Invertibleadditivefunctions 322
12.6Levelsets 327
12.7Partitions 330
12.8Monotonicity 335
Part III Related Topics
13 Related Equations
13.1TheremainingCauchyequations 343
13.2Jensenequation 351
13.3Pexiderequations 355
13.4Multiadditivefunctions 363
13.5Cauchyequationonaninterval 367
13.6TherestrictedCauchyequation 369
13.7 Hossz´uequation 374
13.8 Mikusi´nskiequation 376
13.9Analternativeequation 380
13.10Thegenerallinearequation 382
14 Derivations and Automorphisms
14.1Derivations 391
14.2Extensionsofderivations 394
14.3Relationsbetweenadditivefunctions 399
14.4 Automorphisms of R 402
14.5 Automorphisms of C 403
14.6 Non-trivial endomorphisms of C 406
15 Convex Functions of Higher Orders
15.1Thedifferenceoperator 415
15.2Divideddifferences 421

15.3Convexfunctionsofhigherorder 429
15.4 Local boundedness of p-convexfunctions 432
15.5 Operation H 435
15.6 Continuous p-convexfunctions 439
15.7 Continuous p-convex functions. Case N =1 442
15.8 Differentiability of p-convexfunctions 444
15.9Polynomialfunctions 446
Contents xi
16 Subadditive Functions
16.1Generalproperties 455
16.2 Boundedness. Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 458
16.3 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
16.4Sublinearfunctions 471
16.5Norm 473
16.6 Infinitary subadditive functions . . . . . . . . . . . . . . . . . . . . . . 475
17 Nearly Additive Functions and Nearly Convex Functions
17.1Approximatelyadditivefunctions 483
17.2Approximatelymultiadditivefunctions 485
17.3 Functions with bounded differences . . . . . . . . . . . . . . . . . . . . 486
17.4Approximatelyconvexfunctions 490
17.5Setideals 498
17.6Almostadditivefunctions 505
17.7Almostpolynomialfunctions 510
17.8Almostconvexfunctions 515
17.9Almostsubadditivefunctions 524
18 Extensions of Homomorphisms
18.1Commutativedivisiblegroups 535
18.2 The simplest case of S generating X 537
18.3Ageneralization 540
18.4Furtherextensiontheorems 546

18.5Cauchyequationonacylinder 551
18.6Cauchynucleus 556
18.7TheoremofGer 560
18.8Inverseadditivefunctions 564
18.9Concludingremarks 569
Bibliography 571
Indices
Index of Symbols 587
Subject Index 589
Index of Names 593
Introduction
The present book is based on the course given by the author at the Silesian University
in the academic year 1974/75, entitled Additive Functions and Convex Functions.
Writing it, we have used excellent notes taken by Professor K. Baron.
It may be objected whether an exposition devoted entirely to a single equation
(Cauchy’s Functional Equation) and a single inequality (Jensen’s Inequality) deserves
the name An introduction to the Theory of Functional Equations and Inequalities.
However, the Cauchy equation plays such a prominent role in the theory of functional
equations that the title seemed appropriate. Every adept of the theory of functional
equations should be acquainted with the theory of the Cauchy equation. And a sys-
tematic exposition of the latter is still lacking in the mathematical literature, the
results being scattered over particular papers and books. We hope that the present
book will fill this gap.
The properties of convex functions (i.e., functions fulfilling the Jensen inequality)
resemble so closely those of additive functions (i.e., functions satisfying the Cauchy
equation) that it seemed quite appropriate to speak about the two classes of functions
together.
Even in such a large book it was impossible to cover the whole material pertinent
to the theory of the Cauchy equation and Jensen’s inequality. The exercises at the

end of each chapter and various bibliographical hints will help the reader to pursue
further his studies of the subject if he feels interested in further developments of
the theory. In the theory of convex functions we have concentrated ourselves rather
on this part of the theory which does not require regularity assumptions about the
functions considered. Continuous convex functions are only discussed very briefly in
Chapter 7.
The emphasis in the book lies on the theory. There are essentially no examples
or applications. We hope that the importance and usefulness of convex functions
and additive functions is clear to everybody and requires no advertising. However,
many examples of applications of the Cauchy equation may be found, in particular, in
books Acz´el [5] and Dhombres [68]. Concerning convex functions, numerous examples
are scattered throughout almost the whole literature on mathematical analysis, but
especially the reader is referred to special books on convex functions quoted in 5.3.
We have restricted ourselves to consider additive functions and convex functions
defined in (the whole or subregions of) N-dimensional euclidean space R
N
.Thisgives
the exposition greater uniformity. However, considerable parts of the theory presented
xiv Introduction
can be extended to more general spaces (Banach spaces, topological linear spaces).
Such an approach may be found in some other books (Dhombres [68], Roberts-Varberg
[267]). Only occasionally we consider some functional equations on groups or related
algebraic structures.
We assume that the reader has a basic knowledge of the calculus, theory of
Lebesgue’s measure and integral, algebra, topology and set theory. However, for the
convenience of the reader, in the first part of the book we present such fragments of
those theories which are often left out from the university courses devoted to them.
Also, some parts which are usually included in the university courses of these subjects
are also very shortly treated here in order to fix the notation and terminology.
In the notation we have tried to follow what is generally used in the mathematical

literature
1
. The cardinality of a set A is denoted by card A.Thewordcountable or
denumerable refers to sets whose cardinality is exactly ℵ
0
. The topological closure and
interior of A are denoted by cl A and int A. Some special letters are used to denote
particular sets of numbers. And so N denotes the set of positive integers, whereas
Z denotes the set of all integers. Q stands for the set of all rational numbers, R for
the set of all real numbers, and C for the set of all complex numbers. The letter
N is reserved to denote the dimension of the underlying space. The end of every
proof is marked by the sign . Other symbols are introduced in the text, and for the
convenience of the reader they are gathered in an index at the end of the volume.
The book is divided in chapters, every chapter is divided into sections. When
referring to an earlier formula, we use a three digit notation: (X.Y.Z) means formula
Z in section Y in Chapter X. The same rule applies also to the numbering of theorems
and lemmas. When quoting a section, we use a two digit notation: X.Y means section
Y in Chapter X. The same rule applies also to exercises at the end of each chapter. The
book is also divided in three parts, but this fact has no reflection in the numeration.
Many colleagues from Poland and abroad have helped us with bibliographical
hints and otherwise. We do not endeavour to mention all their names, but nonetheless
we would like to thank them sincerely at this place. But at least two names must be
mentioned: Professor R. Ger, and above all, Professor K. Baron, whose help was
especially substantial, and to whom our debt of gratitude is particularly great. We
thank also the authorities of the Silesian University in Katowice, which agreed to
publish this book. We hope that the mathematical community of the world will find
it useful.
Katowice, July 1979
Marek Kuczma
1

The notation in the second edition has been slightly changed. The following sentences are modified
accordingly.
Part I
Preliminaries
Chapter 1
Set Theory
1.1 Axioms of Set Theory
The present book is based on the Zermelo-Fraenkel system of axioms of the Set
Theory augmented by the axiom of choice. The axiom of choice plays a fundamental
role in the entire book. The mere existence of discontinuous additive functions and
discontinuous convex functions depends on that axiom
1
. Therefore the axiom of choice
will equally be treated with the remaining axioms of the set theory and no special
mention will be made whenever it is used.
The primitive notions of the set theory are: set, belongs to (∈), and being a
relation type (τ)[ατA, R means α is a relation type of A, R; cf. Axiom 8]. The
eight axioms read as follows.
Axiom 1.1.1. Axiom of Extension. Two sets are equal if and only if they have the
same elements:
A = B if and only if (x ∈ A) ⇔ (x ∈ B).
Axiom 1.1.2. Axiom of Empty Set. There exists a set ∅ which does not contain any
element:
For every x, x /∈ ∅.
Axiom 1.1.3. Axiom of Unions. For every collection
2
A of sets there exists a set

A

which contains exactly those elements that belong to at least one set from A:

x ∈

A

⇔ (there exists an A ∈Asuch that x ∈ A).
Axiom 1.1.4. Axiom of Powers. For every set A there exists a collection P(A) of sets
which consists exactly of all the subsets of A:

B ∈P(A)

⇔ (B ⊂ A).
1
R. M. Solovay has shown (Solovay [292]) that a model of mathematics (without axiom of choice)
is possible in which all subsets of R (and consequently also all functions f : R → R)areLebesgue
measurable.
2
The word collection is, of course, a synonym of set.
4 Chapter 1. Set Theory
Axiom 1.1.5. Axiom of Infinity. There exists a collection A of sets which contains the
empty set ∅ and for every X ∈Athere exists a Y ∈Aconsisting of all the elements
of X and X itself:
∅ ∈Aand for every X ∈Athere exists a Y ∈Asuch that
(x ∈ Y ) ⇔ (x ∈ X) or (x = X).
Axiom 1.1.6. Axiom of Choice. The cartesian product of a non-empty family of non-
empty sets is non-empty:
If A = ∅ and for every A ∈A,A= ∅, then ×
A∈A
A = ∅.

Axiom 1.1.7. Axiom of Replacement. Let ψ(x, y) be a two-place propositional formula
such that, for every x, there exists exactly one y such that ψ(x, y) holds. Then for
every set A there exists a set B which contains those and only those y for which
ψ(x, y),x∈ A:
If for every x there exists a z such that ψ(x, y) ⇔ y = z, then for every set A
there exists a set B such that
(y ∈ B) ⇔ there exists an x ∈ A such that ψ(x, y).
Roughly speaking, if to every x there corresponds (according to ψ) a unique y,
and if x runs over a set, then the corresponding y’s run over a set.
Before stating the last axiom, we must introduce certain notions. Let A be a
set and R ⊂ A
2
arelationinA. A couple A, R is called a relation system. Two
relation systems A, R and B, S are said to be isomorphic iff there exists a one-to-
one function f from A onto B (a bijection) such that for every a, b ∈ A we have aRb
if and only if f(a)Sf(b).
Axiom 1.1.8. Axiom of Relation Types. To every relation system A, R there corre-
sponds an object α such that ατA, R and if ατA, R and βτB,S,thenα = β if
and only if A, R and B,S are isomorphic.
The Axiom of Replacement implies the following statement.
Axiom of Specification. For every set A and for every propositional formula Φ(x)
there exists the set {x ∈ A | Φ(x)} consisting of exactly those x ∈ A for which Φ(x
)
holds true:
x ∈{x ∈ A | Φ(x)}⇔x ∈ A ∧ Φ(x).
It is enough to take ψ(x, y) ⇔ Φ(x) ∧ y = x.
The Axiom of Empty Set can be replaced by the weaker
Axiom of Existence. There exists a set.
Theemptysetcanbedefinedas(A being an existing set)
∅ = {x ∈ A | x = x}.

1.2. Ordered sets 5
If we take into account the definition of the cartesian product of an arbitrary
collection of sets, we can reformulate the Axiom of Choice as follows:
For every non-empty collection A of non-empty sets there exists a function w :
A→

A (the choice function) such that w(A) ∈ A for every A ∈A.
The Axiom of Choice is usually used in this form.
The Axiom of Extension implies the uniqueness of sets whose existence is guar-
anteed by the remaining Axioms 2–7.
The Axiom of Relation Types can be omitted. The whole set theory can be built
without a use of this axiom. The ordinal numbers (as well as cardinal numbers) must
then be defined otherwise. (Cf., e.g., Halmos [130]).
From the Axioms 1.1.1–1.1.8 all the set theory can be built (cf. Kuratowski-
Mostovski [198], Halmos [130], Rasiowa [262]). We assume that the reader is familiar
with it. However, in the sequel we outline the theory of ordinal numbers, as the latter
is often omitted in the university courses of the set theory.
1.2 Ordered sets
Let A be a set, and  ⊂ A
2
a relation which is reflexive, antisymmetric and transitive:
(i) a  a,
(ii)(a  b) ∧ (b  a) ⇒ (a = b),
(iii)(a  b) ∧ (b  c) ⇒ (a  c).
Such a relation  is called an order
3
in A and the couple (A, ) is called an ordered
set. Clearly, every ordered set is a relation system in the sense of 1.1.
The strict relation < is defined as
(a<b) ⇔ (a  b) ∧(a = b).

Instead of a  b, a<b, we shall often write b  a, b>a.
If, besides (i), (ii)and(iii), also the trichotomy law holds:
(iv) For every a, b ∈ A,wehaveeithera<b,orb<aor a = b, then the set A is
called linearly ordered or a chain.
Let (A, ) be an ordered set. An element a ∈ A is called maximal [minimal]iff
there is no b ∈ A strictly greater [smaller] than a.Inotherwords,a is maximal iff
(b ∈ A) ∧ (a  b
) ⇒ (b = a).
[a is minimal iff
(b ∈ A) ∧ (b  a) ⇒ (b = a)].
3
In earlier texts order is often called a partial order,thewordorder being reserved for what is here
called a linear order. This is due to the fact that, for arbitrary a, b ∈ A, we are often unable to
decide whether a  b or b  a. An illustrative example is the power set P(A)ofasetA with the
order relation defined as the inclusion:
a  b ⇔ a ⊂ b.
6 Chapter 1. Set Theory
One ordered set may have several (or none) maximal [minimal] elements. If
a ∈ A is a maximal [minimal] element, then there may exist in A elements b which
are not comparable with a, i.e., for which neither a  b,norb  a holds.
An element a ∈ A is called the greatest [smallest](orthelast [least]) element iff
x  a [a  x]holdsforeveryx ∈ A. The last [least] element, if it exists, is unique.
An element a ∈ A is called the upper bound of a set E ⊂ A iff x  a holds for
every x ∈ E. It is not required that a ∈ E, but it is possible. There may exist several
(or none) upper bounds of a set E ⊂ A.
If (B,) is another ordered set, then we say that (A, )and(B,)aresimilar

and write (A, ) ∼ (B,≺)

iff there exists a one-to-one order-preserving mapping

f from A onto
B. The relation of similarity is an isomorphism of relation systems
(A, )and(B, ) as defined in 1.1:
(a  b) ⇔

f(a)  f(b)

An ordered set, every non-empty subset of which has the smallest element, is
called a well-ordered set, and the corresponding order is called a well-order.Wehave
the following
Theorem 1.2.1. Every well-ordered set is linearly ordered.
Proof. This follows from the fact that for any a, b ∈ A, the pair {a, b}⊂A has the
smallest element. 
Any finite linearly ordered set is well ordered and two such sets are similar if and
only if they have the same number of elements. (The proof of these facts is left to the
reader.) The set (N, ), where  stands for the usual inequality between numbers,
is well ordered.
1.3 Ordinal numbers
Let (A, ) be a well-ordered set. Any set P ⊂ A such that if x ∈ P and y  x,then
y ∈ P , is called an initial segment of A.
Theorem 1.3.1. If P is an initial segment of a well-ordered set A
4
,andP = A,then
there exists in A an x such that
P = P (x)={y ∈ A | y<x}.
Proof. The set A \P = ∅ has the smallest element x. We will show that P = P (x).
Let y ∈ P .Ifwehadx  y, then we would have x ∈ P , which contradicts the
condition x ∈ A \ P.Thusy<xand y ∈ P (x). Consequently P ⊂ P (x).
If y ∈ P (x), then y<x, and since x is smallest in A \ P ,wemusthavey ∈ P .
Consequently P(x) ⊂ P .

Thus P and P (x) have the same elements, and so they are equal: P = P (x). 
4
Instead of saying: A is the first component of the ordered set (A, ), we often say simply: A is an
ordered set.
1.3. Ordinal numbers 7
The formulas, valid for arbitrary a, b ∈ A, are left to the reader as exercises:
1. (a  b) ⇔

P (a) ⊂ P (b)

,
2. (a = b) ⇔

P (a) = P (b)

,
3. For arbitrary well-ordered set (A, ) put P = {P(x) | x ∈ A}. The ordered set
(P, ⊂) is well ordered and is similar to (A, ).
The relation types of well-ordered sets are called the ordinal numbers.If(A, )
is a well-ordered set and if ατ (A, ), then we write α =
A.TheN is denoted by ω.
If (A,  ) is a finite (well-ordered) set consisting of n ∈ N elements, then we assume
A = n.Inparticular,∅ =0.
If (A, ) is a well-ordered set, then, by the Axiom of Infinity there exists a
set B which contains all the elements of A and A itself. We order B by assuming
additionally that A  a for any a ∈ A. The ordinal number B is denoted by A +1. If
an ordinal number cannot be written as α + 1 with another ordinal number α,then
it is called a limit number. An example of a limit number is ω.
If α and β are ordinal numbers, say α =
A and β = B,thenwesaythatα<β

iff the set A is similar to an initial segment of B different from B.
Theorem 1.3.2. For any ordinal number α, it is not true that α<α.
Proof. For an indirect proof, suppose that α<α, i.e., A is similar to its initial
segment different from A.Letf be a similarity function. Put
B = {x ∈ A | f(x) <x}.
By Theorem 1.3.1 there exist an a ∈ A such that A ∼ P (a). Hence f(a) <a, i.e.,
a ∈ B.SoB = ∅,andB ⊂ A, and hence there exists the smallest element, say b,in
B.Then
f(b) <b, (1.3.1)
and since f is order-preserving, f

f(b)

<f(b). This means that f(b) ∈ B,which,by
(1.3.1) contradicts the condition that b is smallest in B. 
Theorem 1.3.3. If α<βand β<γ,thenα<γ.
Proof. Let A, B, C be well ordered sets such that
A = α, B = β and C = γ.Moreover,
let b ∈ B and c ∈ C be such that A ∼ P (b)andB ∼ P (c), the similarity functions
being f and g, respectively. Then it is easily checked that g ◦ f is a one-to-one,
order-preserving mapping of A onto an initial segment of C, different from C. 
Theorem 1.3.4. If α<β, then it is not true that β<α.
Proof. This follows from Theorems 1.3.3 and 1.3.2. 
Theorems 1.3.2 and 1.3.3 imply that the inequality  defined for ordinal numbers
as follows: α  β iff either α<β,orα = β, is an order in the sense of 1.2.
8 Chapter 1. Set Theory
1.4 Sets of ordinal numbers
We start with a lemma.
Lemma 1.4.1. If (A, ) and (B, ) are similar well-ordered sets
5

and f is a similarity
function, then f maps initial segments of A onto initial segments of B.
Proof. Let f be a one-to-one order preserving mapping of A onto B.Thenf
−1
is a
one-to-one order preserving mapping of B onto A.LetP be an initial segment of A.
The thing to prove is that f(P ) is an initial segment of B.
Suppose that there exist b
1
<b
2
, b
2
∈ f(P )andb
1
/∈ f(P ). Put a
i
= f
−1
(b
i
),
i =1, 2. Then a
1
<a
2
, a
2
∈ P and a
1

/∈ P , which is impossible since P is an initial
segment of A. 
Corollary 1.4.1. No two different initial segments of a well-ordered set are similar to
each other.
Proof. Let (A, ) be a well-ordered set, and P
1
= P
2
initial segments of A, P
1
∼ P
2
.
Since P
1
= P
2
, at least one of these segments, say P
1
, must be different from A,and
hence of the form P (a)withana ∈ A.Ifa ∈ P
2
,thenP
1
is an initial segment of
(P
2
, ). Indeed, if x ∈ P
1
and y<x,theny<aand hence y ∈ P (a)=P

1
.Andif
a/∈ P
2
,thenP
2
is an initial segment of (P
1
, ). Indeed, then P
2
= A and hence of
the form P (b)withab ∈ A.Ifx ∈ P
2
and y<x,theny<band y ∈ P
2
.
Thus one of the sets P
1
,P
2
is similar to its initial segment different from this
set, which contradicts Theorem 1.3.2. 
Theorem 1.4.1. Of any two well-ordered sets, one is similar to an initial segment of
the other.
Proof. Let (A, )and(B, ) be two well-ordered sets. Define the set Z,
Z = {x ∈ A | there exists a y ∈ B such that P (x) ∼ P (y)}.
By Corollary 1.4.1, such a y is unique. Thus we may define a function f : Z → B by
putting f(x)=y iff P (x) ∼ P (y). Again by Corollary 1.4.1 f is one-to-one.
The set Z is an initial segment of A.Forifx ∈ Z,thenP (x) ∼ P (y)fora
y ∈ B.Ifx


<x,thenP (x

) ⊂ P(x). Let g, mapping P (x)ontoP (y), be a similarity
function. Then g maps P(x

) onto an initial segment P of P(y) and hence P is an
initial segment of B.Sincey/∈ P (y), also y/∈ P , and thus P = B.Sothereexistsa
y

∈ B such that P = P (y

). Hence g establishes a similarity between P (x

)andP (y

).
Thus x

∈ Z and P (y

)=P

f(x

)

is an initial segment of P (y)=P

f(x)


, whence
f(x

) <f(x). So as a by-product we have obtained the fact that f is order-preserving.
Similarly, f (Z) is an initial segment of B.Forify ∈ f(Z), then P (y) ∼ P (x)
for an x ∈ Z.Leth be a similarity mapping. If y

<y,thenh maps P (y

)ontoan
initial segment P (x

)ofP (x), whence y

= f (x

), x

∈ Z and y

∈ f (Z).
5
It would be enough to postulate that one of these sets is well ordered. It follows then by the
similarity that the other is well ordered, too. (Cf. Exercise 1.6)
1.4. Sets of ordinal numbers 9
We have already shown that Z ∼ f (Z). To complete the proof it is enough to
show that either Z = A or f (Z)=B. Suppose that Z = A and f(Z) = B.Then
there exist a ∈ A and b ∈ B such that Z = P (a)andf(Z)=P(b). Thus P (a) ∼ P (b)
and a ∈ Z, which is incompatible with Z = P (a). 

Theorem 1.4.1 implies the trichotomy law for ordinal numbers:
For any ordinal numbers α, β
either α<β,orβ<α,orα = β.
Let us note also the following
Theorem 1.4.2. For every ordinal number α, there exists the set Γ(α) of all ordinal
numbers β<α.
Proof. Let (A, ) be a well-ordered set such that
A = α.Letforx ∈ A, ψ(x, y)be
the propositional formula yτP(x), i.e., y =
P (x). By the Axiom of Replacement there
exists a set B such that y ∈ B ⇔ there exists an x ∈ A such that yτP(x). But this,
in turn, is equivalent to the fact that y<α.ThusB is the required set. 
Once we know that Γ(α) is a set, the formula y =
P (x)forx ∈ A defines a
function from A onto Γ(α). This function clearly is one-to-one and order-preserving.
It follows that Γ(α) ∼ A and consequently Γ(α) is well ordered and
Γ(α)=α. (1.4.1)
Actually, the fact that Γ(α) is well ordered is a particular case of the following
statement.
Theorem 1.4.3. Any set of ordinal numbers is well ordered by the inequality .
Proof. The thing to prove is that if A = ∅ is a set of ordinal numbers, then there
exists the smallest element in A.Takeanα ∈ A.Ifα is smallest in A, there is nothing
more to prove. If not, the set A ∩Γ(α) ⊂ Γ(α) = ∅.SinceΓ(α) is well ordered, there
exists the smallest element β in A ∩ Γ(α). For an arbitrary γ ∈ A we have either
γ  α,orγ<α. In the first case, since β ∈ Γ(α), we have β<α γ, whence β<γ.
In the other case γ ∈ Γ(α) ∩A,soβ  γ.Thusβ is the smallest element in A. 
Next we have
Lemma 1.4.2. For every ordinal number α we have α<α+1.
Proof.
Let α =

A.Thenα+1 = A
∗
,whereA
∗
is the set consisting of all the elements
of A and A itself ordered so that A  a for all a ∈ A. Hence P (A)=A in A
∗
.The
function f : A → A
∗
defined as f(a)=a for a ∈ A establishes the similarity of A
with P(A)inA
∗
.ThusA is similar to an initial segment of A
∗
different from A
∗
, i.e.,
A<A
∗
. 
Theorem 1.4.4. For every set of ordinal numbers there exists an ordinal number
strictly greater than any number from the given set.
10 Chapter 1. Set Theory
Proof. Let A be a set of ordinal numbers. By the Axiom of Replacement there exists
the collection of sets {Γ(β)}
β∈A
, and by the Axiom of Unions there exists the set
B =


β∈A
Γ(β).
Then, for every β ∈ A,thesetΓ(β) clearly is an initial segment of B. Hence
B 
Γ(β)=β

cf. (1.4.1)

. In view of Lemma 1.4.2 the ordinal number B + 1 has the
desired property. 
Corollary 1.4.2. There does not exist the set of all ordinal numbers.
Corollary 1.4.3. For every set of ordinal numbers there exists the smallest ordinal
number which does not belong to the given set.
Proof. Let A be a set of ordinal numbers and let α be a number greater than any
β ∈ A.Ifα is not the smallest number with this property, then the set Γ(α)\A ⊂ Γ(α)
is non-empty, and consequently it has the smallest element γ. It is already seen that
γ is the smallest ordinal number which does not belong to A. 
1.5 Cardinality of ordinal numbers
If α is an ordinal number, then by definition any two well-ordered sets of type α are
similar, i.e., there exists a one-to-one mapping from one set onto the other. Conse-
quently these sets have the same cardinality. Consequently to any ordinal number α
we may assign a cardinal number, the common cardinality of all well-ordered sets of
type α. This cardinal number is called the cardinality of α and is denoted by
α.In
particular we have

cf. (1.4.1)

α =cardΓ(α).
It is easy to check that for arbitrary ordinal numbers α, β we have

α  β ⇒
α  β, (1.5.1)
Lemma 1.5.1. There exists the set A which contains all ordinal numbers α with
α  ℵ
0
.
Proof. If
α  ℵ
0
for an ordinal number α, then there exists a one-to-one mapping
f :Γ(α) → N.Iffora, b ∈ f

Γ(α)

we put
(a  b) ⇔

f
−1
(a)  f
−1
(b)

,
the set

f

Γ(α)


, 

will become a well-ordered set similar to Γ(α)(f being the
similarity function), and hence

f

Γ(α)

, 

= α. Thus for any ordinal number α
with
α  ℵ
0
there exists a well-ordered set (B, ) such that B ⊂ N,andB = α.The
converse is also true. If (B, ) is a well-ordered set of a type α,andB ⊂ N,then
α  ℵ
0
. Consequently we may describe ordinal numbers α such that α  ℵ
0
as order
1.5. Cardinality of ordinal numbers 11
types of well-ordered subsets of N. (However, the order in these sets may be different
from the natural order in N.)
Let P be the power set of N (Axiom 4) and, for P ∈P,letR
P
be the power set
of P ×P . Every element R of R
P

is a relation in P . There exists the set (Axiom of
Specification)
B = {(P, R) | P ∈P,R ∈R
P
}
and the set
C = {(P, R) ∈ B | R is a well order}.
By the Axiom of Replacement there exists the set of the types of all (P, R) ∈ C.This
is the desired set. 
The same argument shows that for any cardinal number m there exists the set
of all ordinal numbers α such that α  m . This set is denoted in the sequel by M(m).
Lemma 1.5.2. For every cardinal number m, we have
M(m)=Γ

M(m)

. (1.5.2)
Proof. There exists an ordinal number β which is greater than any α ∈ M(m). First
we show that M (m) is an initial segment of Γ(β).
Let γ ∈ M(m)andξ<γ. Then by (1.5.1)
ξ  γ  m, i.e., ξ ∈ M(m). So either
M(m)=Γ(β), or there exists an η ∈ Γ(β) such that
M(m)={ξ ∈ Γ(β) | ξ<η} =Γ(η).
At any case M(m)=Γ(α) for an ordinal number α.Sinceα =
Γ(α)=M(m), we
obtain hence (1.5.2). 
Theorem 1.5.1. For every cardinal number m, we have
card M(m) > m. (1.5.3)
Proof. Put α =
M(m)sothatM(m)=Γ(α). Suppose that (1.5.3) does not hold,

i.e.,
α  m. This means that α ∈ M(m), i.e., α ∈ Γ(α), which is impossible. 
Remark 1.5.1. Here we have made use of the trichotomy law for the cardinals, which,
however, cannot be proved at this stage. What we actually prove here is that the
inequality
α  m is impossible. This is sufficient to prove the remaining theorems
of the present chapter, and then the trichotomy law for the cardinals follows from
Theorem 1.4.1 and 1.7.1, and hence also condition (1.5.3).
We define Ω to be the order type of the set M(ℵ
0
), and ℵ
1
to be the cardinality
of M(ℵ
0
). Thus
Ω=
M(ℵ
0
), ℵ
1
= Ω=cardM (ℵ
0
).
By Theorem 1.5.1 ℵ
1
> ℵ
0
.Moreover
Theorem 1.5.2. There is no cardinal number strictly between ℵ

0
and ℵ
1
.
In other words, ℵ
1
is the next cardinal number after ℵ
0
.