[Slfm 118] theory of relations roland fraisse (nh 1986)(t)

STUDIESINLO-GIC
AND
THE FOUNDATIONS OF MATHEMATICS
VOLUME 118
Editors
J. BARWISE, Stanford
D. KAPLAN, LosAngeles
H. J . KEISLER, Madison
P. SUPPES,Stanford
A. S.TROELSTRA.Amsterdam
NORTH-HOLLAND
AMSTERDAM NEW YORK OXFORD

THEORY
OF
RELAmONS
R.FRAISSE
Universitk de Provence
Marseille
France
1986
NORTH-HOLLAND
AMSTERDAM 0 NEWYORK OXFORD

ELSEVIER SCIENCE PUBLISHERS B.V.,1986
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F r a i s d , Roland.
Theory of relations.
(Studies i n logic a d the foundations of
matheratics ; v. 118)
Translation of: Thhrie des relations.
Bibliography: p.
Includes index.
1. Set theory. I. Title. 11. Series.
QA248.FT75 1986 511.3'22 85-20701
ISBN 0-444-87865-3
PRINTED IN THE NETHERLANDS

V
INTRODUCTION
Relation theory goes back t o the 1940's ; i t originates i n the theory o f order
types, due t o HAUSDORFF (Grundzuge der Mengenlehre 1914), SIERPINSKI (Le~onssur
les nombres t r a n s f i n i s 1928, taken up again i n Cardinal and ordinal numbers 1958),
SZPILRAJN (Sur l'extension de l ' o r d r e p a r t i e l 1930), DUSHNIK, MILLER (Concerning
s i m i l a r i t y transformations of 1inearly ordered sets 1940), GLEYZAL (Order types
and structure o f orders 1940), and t o HESSENBERG (Grundbegriffe der Mengenlehre
1906, introducing the negative and rational ordinals). A t t h a t time, r e l a t i o n
theory j u s t extended t o a r b i t r a r y relations the elementary notions o f order type
and embeddability.
Relation theory intersects only weakly with graph theory, w i t h which i t i s
sometimes s t i l l confused. F i r s t l y , echniques i n r e l a t i o n theory only rarely
distinguish between graphs, i.e. s j m t r i c binary relations, and relations o f
arbitrary a r i t y . Additionally, as opposed t o graph theory, i n r e l a t i o n theory
one considers equally the two t r u t h values (+) and (-) taken on by a r e l a t i o n
with base E f o r each element o f E2 ( o r o f En f o r the a r i t y n ).
On the other hand, r e l a t i o n theory uses techniques especially from combinatorics,
the l a t t e r which can be defined as f i n i t e set theory. Anything concerning relations
with f i n i t e bases, or counting isomorphism types o f f i n i t e r e s t r i c t i o n s o f a given
relation, o r again the study o f permutations o f the base which preserves a given
relation ( i .e. automorphisms o f the relation), makes use o f combinatorics. From
a more technical viewpoint, see the combinatorial lemnas i n ch.3 5 4, and the
study o f the incidence matrix i n ch.3 5 5.
' a
As f o r mathematical logic, i t s intersection w i t h r e l a t i o n theory i s rather impor-
tant. One can even say t h a t the two principal sources f o r r e l a t i o n theory are
the study o f order types, already mentioned, and l i n e a r logic, i.?. first-order
one-quantifier logic; t h a t i s the study o f universal formOlas (prenex formulas
only having universal quantifiers), and boolean combinations thereof, with the
particular case o f quantifier-free formulas. From a semantic, o r model-theoretic
viewpoint, t h i s i s the study o f universal classes o f TARSKI,' VAUGHT 1953, and
o f boolean combinations thereof.
If one presents mathematical l o g i c from a relational theoretic viewpoint, the
basic notion i s t h a t o f local isomorphism, i.e. isomorphism o f a r e s t r i c t i o n o f
the f i r s t r e l a t i o n onto a r e s t r i c t i o n o f the second one: see ch.9 5 1.4. For
example, the free i n t e r p r e t a b i l i t y o f a r e l a t i o n S i n another r e l a t i o n R
with the same base, i s algebrically defined by the condition that every local
automorphism o f R (local isomorphism from R i n t o R ) i s also a local auto-
morphism o f S . Equivalently, free i n t e r p r e t a b i l i t y i s l o g i c a l l y defined by
the existence o f a quantifier-free formula which defines S i n the structure
of R . For example, i f R i s a chain, o r t o t a l ordering, then the betweenness
r e l a t i o n S(x,y,z) = + i f f z i s between x and y , i s defined by the
quantifier-free formula (Rxz h Rzy) v (Ryz A Rzx) . This equivalence between
algebraic and l o g i c a l notions exists even above the free-quantifier and the
one-quantifier cases; since logical ( o r elementary) equivalence between R
and S , saying t h a t R and S satisfy the same f i r s t - o r d e r formulas, i s
equivalent t o t h e i r being (k,p)-equivalent f o r a l l integers k and p ,
which i s a purely algebraic notion: see my Course o f mathematical l o g i c 1974
vol. 2 . Coming back t o the l i n e a r case (one-quantifier logic), as common notions
and techniques i n both mathematical logic and r e l a t i o n theory, we have those o f
1-isomorphism, 1-extension, projection f i l t e r ( a variant o f ultraproduct): see
ch.10 5 1 . And f o r each ordinal o( , the 4 -morphism (ch.10 5 4), which i s not

vi Introduction
a one-quantifier notion, b u t i s indispensible in relation theory for the study of
embeddability: see ch. 10 5 4 and 5 5.3.
From the 1970's, an important connection appears between relation theory and the
theory of permutations. See the study of orbits (ch.11 § 2 ) , the theorem on the
increasing number of orbits (LIVINGSTONE, WAGNER, ch.11 5 2.8) and the theorem on
set-transitive, or homogeneous groups (CAMERON, ch.11 5 5.10).
Let us mention, also from the 1970's, some unexpected connections between relation
theory and topology (ch.1 5 8 and ch.7 5 2 ) ; and even connections w i t h linear
algebra (ch.11 5 2.6).
We shall now briefly present the principal notions studied, by mentioning f i r s t
that chapters 1 through 8 concern the theory of partial and total orderings (or
chains), while chapters 9 through 12 concern the general study of relations.
In chapter 1, we review basic set theoretical results, in general without proofs,
which allow the reader t o know, for instance, i n which precise sense we use the
notion of f i n i t e set (TARSKI's sense rather than OEDEKIND's), or the notion of
cardinality of a set. This allows us t o precise, throughout the r e s t of the book,
which axioms are used for each proof: ZF alone, the axiom of choice, dependent
choice, the u l t r a f i l t e r axiom, the continuum hypothesis, etc. Moreover i t seems
that even among logicians, there are few who are aware that, while O1 > W
is provable in ZF alone, yet the countable axiom of choice, for instance, is used
t o prove that W 1 i s regular. Or that KONIG's theorem (ch.1 0 1.8), even in
the very particular case of two ordered pairs of s e t s , i s not provable in ZF alone.
Or that the possible equivalence between the axiom o f choice and the statement
that the range of a function is subpotent w i t h its domain, i s s t i l l an open
problem, already p u t forth by RUBIN 1963. Thus this chapter could be useful as a
memory brush-up for the axiomatic s e t theoretician.
In chapter 2 , in addition t o a review of basic relation theoretical notions,
similar in s p i r i t t o chapter 1, we introduce some notions which are no longer
classical, yet which extend well-known concepts. For example the coherence lemma
( 5 1.3), a not well-known version of the u l t r a f i l t e r axiom. Another example, the
cofinality of a partial ordering, as well as the related notion of cofinal height
(5 5.4 and 5 7). Classically, the notion of cofinality is relegated t o the single
case of chains, or total orderings, which while interesting is too much restric-
tive.
In chapter 3, we present RAMSEY's theorem and important refinements of i t , due t o
GALVIN and t o NASH-WILLIAMS (5 2 ) . Furthermore, the " i n i t i a l interval theorem"
or GALVIN's theorem is presented twice, with very different proofs: POUZET's
proof i n 5 2 and LOPEZ'S proof using the classical Ramsey sets of reals, in 5 6.
Then we are led t o the partition theorems of DUSHNIK, MILLER and of ERDOS, RADO.
We also present a combinatorial study of the incidence matrix, w i t h the linear
independence lemma due t o KANTOR.
In chapter 4, we begin the study of partial orderings, w i t h the notions of good
and bad sequence, of a f i n i t e l y free partial ordering, and that of a well partial
ordering. We present HIGMAN's characterization of a well partial ordering (the
s e t of i n i t i a l intervals is well-founded under inclusion); also HIGMAN's theorem
on words i n a well partial ordering, and RADO's well partial ordering (5 4 ) . Also
the notions of ideal, tree, dimension, bound of an i n i t i a l interval. We present
the theorem of the maximal reinforced chain for a well partial ordering, due t o
DE JONGH, PARIKH (5 9 ) . The chapter ends ( 5 10) w i t h POUZET's theorem on regular
(or f i n i t e ) cofinality of any finitely free partial ordering.

Introduction vii
In chapter 5, we consider embeddability between orderings, the well partial orde-
ring of finite trees (KRUSKAL), the existence of immediate extensions and of
faithful extensions (HAGENDORF), Cantor's theorem for partial orderings (DILWORTH,
GLEASON). Then the existence of strictly decreasing infinite sequences of chains
of reals: the denumerable sequence due to DUSHNIK, MILLER and the continuum
length sequence due t o SIERPINSKI. Finally a brief study of SUSLIN's chain and
tree, in connection with SUSLIN hypothesis; also ARONSZAJN tree, SPECKER chain.
In chapter 6, we introduce the scattered chain, which does not admit any embedding
of the chain Q of the rationals. Also the indecomposable, as well as the right
and the l e ft indecomposable chain. We present HAGENDORF's theorem of unique decom-
position of an indecomposable chain ( 5 3.3) and some connected results (JULLIEN,
LARSON). We begin t o study the covering of a chain by r l g h t or le ft indecomposable
intervals, or by doublets of indecomposable intervals. We present the hereditarily
indecomposable chain with LAVER'S results, and finally the indivisible relation
or chain.
In chapter 7 , we proof supplementary results about finitely free partial orderinas
and their reinforcements by chains. We extend t o the set of initial intervals
the topology already introduced in ch.1 5 8, and give some applications, namely
BONNET'S results. Then we prove the following important theorem of POUZET: every
directed well partial ordering has a cofinal restriction which i s a direct
product of finitely many distinct regular alephs. The chapter ends with a short
study of Szpilrajn chains (BONNET, JULLIEN); two interesting results due t o
TUKEY and t o KRASNER are presented as exercises.
In chapter 8, we introduce the important notion of barrier due to NASH-WILLIAMS;
the partition theorem ( 5 1.4), the theorem on the minimal bad barrier sequence
( 5 2.2); the forerunner and successor barrier. This is the main tool i n the
proof of the very important theorem of LAVER: every set of scattered chains
forms a well quasi-ordering under embeddability (5 4.4). In other words, there
exists neither an infinite strictly decreasing sequence nor an infinite set of
mutually incomparable scattered chains. LAVER proved even more, in extending his
result to chains formed from a countable union of scattered chains. However his
proof has not yet been sufficiently simplified to be presented in a textbook of
a reasonable size. In this chapter, we also study the better partial ordering,
a notion due t o NASH-WILLIAMS, both for i t s intrinsic interest and for i t s
applications to chains.
In chapter 9, we begin the general theory of relations, w i t h the notions of local
isomorphism, free interpretability and free operator (which is the relationist
version of a logical free formula, and links relation theory to logic). We study
constant, chainable, monomorphic relations. In the case of a binary relation with
cardinality p , we present the deep result due to JEAN: (~-2)-monomorphy
implies (p-1)-monomorphy ( 5 6.7). We present the profile increase theorem
(POUZET, 5 7). Finally we extend t o arbitrary relations the homomorphic image
(5 8), and in 5 9 we introduce the bivalent table, which apparently yields
difficult problems, one of them being very partially solved by LOPEZ. Most of
relationist researchers seem to be discouraged by this branch of relation theory,
which i s s t i l l a marginal study inside relation theory, considered itself as
being marginal during too long a time.
In chapter 10, we classify relations according to their age: two representatives
of the same age have the same finite restrictions, up to isomorphism. This i s
equivalent t o classifying relations by the set of universal formulas which they
satisfy. Then we study maximalist or existentially closed relations (3 3.8),
rich relations, inexhaustible relations (5 4 and 5), and relations which are
rich for their age. This notion, connected t o saturated relations, leads t o the
existence criterion of POUZET, VAUGHT (5 7). The fin itist and almost chainable
relations are presented in 5 8 and 9.

viii Introduction
Chapter 11 i s concerned with correspondence between r e l a t i o n theory and permuta-
tions, the l i n k between them being the homogeneous relations and relational
systems. We already mentioned the theorem o f increasing number o f orbits, due
t o LIVINGSTONE, WAGNER. I n 5 3 and following, we introduce the compatibility
modulo a permutation group, which yields a marginal study inside permutation group
theory, with many open problems. The n o t i o m o f indicative group and indicator lead
t o FRASNAY's reduction theorem (5 4). The p a r t i c u l a r case o f Q-indicative groups
leading t o the set-transitive group theorem o f CAMERON. F i n a l l y we study the
pseudo-homogeneous relations, the prehomogeneous relations with POUZET's existence
criterion (5 7), the set-homogeneous relations.
I n chapter 12, we introduce the bounds o f a r e l a t i o n R : f i n i t e relations non-
embeddable i n R but whose proper r e s t r i c t i o n s are embeddable i n R . We present
several important theorems due t o FRASNAY: the reassembling theorem (5 3); the
existence o f an integer p such that, from t h i s point on, p-monomorphism implies
chainability; and the finiteness o f the number o f bounds f o r a chainable relation.
This study uses the method o f permuted chains, o r compatibility modulo a permuta-
t i o n group, already presented i n chapter 11. Proofs have been s i m p l i f i e d by using,
as a powerful tool, the p-well r e l a t i o n due t o POUZET. The chapter, and the book,
are ending with the study o f reduction, reassembling, monomorphism and chainabi-
l i t y thresholds: calculated f i r s t by FRASNAY, they were improved by HODGES,
LACHLAN, SHELAH, then proved again by FRASNAY t o have the smallest possible value.
I n 5 6 we added some easy considerations about universal classes.
I n order t o keep t h i s book t o a reasonable size, we suppressed two planned
chapters. One about the celebrated problem o f reconstruction, i.e. the problem
t o know i n what cases a r e l a t i o n with base E i s completely determined, up
t o isomorphism, by the isomorphism types o f i t s r e s t r i c t i o n s t o proper subsets
o f E . The reader may consult BONDY, HEMMINGER 1977, LOPEZ 1978, 1982, 1983,
POUZET 1979', STOCKMEYER 1977, ULAM 1960 (see Bibliography). The othgr yissing
chapter concerned the notion o f i n t e r v a l i n r e l a t i o n theory: see FRAISSE 1984
i n Bibliography.
Iwould l i k e t o thank those among my colleagues - professors, researchers,
students and ex-students - who solved o r contributed t o the solution o f a l l
problems presented here; and t o those who, by simplifying the inordinately long
or d i f f i c u l t proof o f the o r i g i n a l paper, have made these results accessible,
hence suitable f o r presentation i n t h i s textbook. Their names are mentioned
together w i t h t h e i r contribution. As f o r myself, I have the free conscience o f
having accomplished my work as "chef d'ecole": namely the presentation, i n a form
accessible t o a wide audience, o f results obtained by those who loved my research
area and surpassed me.

1
CHAPTER 1
REVIEW OF AXIOMATIC SET THEORY
The purpose o f t h i s chapter i s t o situate precisely "theory o f relations" within
the framework o f axiomatic set theory, which i n i t i a l l y w i l l be t h a t o f ZERMELO-
FRAENKEL. The axioms f o r ZF are introduced below i n 91 and 92. Our i n i t i a l
notation w i l l be introduced there. I n r e f e r r i n g t o the f i r s t and sometimes
second chapter, we w i l l indicate throughout the book which statements require
only the axioms o f ZF and those which require, t o our knowledge, the axiom o f
choice, o r rather the weaker u l t r a f i l t e r axiom (boolean prime ideal axiom), or
the axiom o f dependent choice, etc. Most o f the proofs, as well as classical
definitions from the f i r s t and second chapter, are l e f t t o the reader.
§ 1 - FIRST GROUP OF AXIOMS FOR ZF, FINITESET, AXIOM OF CHOICE,
KONIG'S THEOREM
We begin with the axioms of extensionality, pair, union, power set (set o f a l l
subsets o f a set) and the scheme o f separation, a l l supposedly known t o the
reader. We denote the empty set by 0 , inclusion C , s t r i c t inclusion C .
We denote the union o f the set a by u a , and the power set by p ( a ) .
I f b s a , we designate the difference by a-b . Singletons, unordered pairs
(simply called pairs) are denoted by a } , a,b) , etc. The successor set
a v { a ) o f a i s denoted by a+l . So t h a t 1 = 0+1 = { 0) i s the successor
o f the empty set; 2 = 1+1 = .( 0,l) i s the successor o f 1 , etc. This nota-
t i o n coincides with the notation f o r ordinal addition, introduced i n fj 3 below.
-
1.1. FINITE SET
Following TARSKI 1924', we define a set a t o be f i n i t e i f f every non-empty
set b o f subsets o f a contains an element which i s minimal with respect t o
inclusion, i.e. an element c E b such t h a t no x E b s a t i s f i e s x c c .
Taking complements, i t i s equivalent t o say t h a t a i s f i n i t e exactly when
every non-empty set o f subsets o f a contains a maximal element. A non-finite
set i s said t o be i n f i n i t e .
The empty set, a singleton, a p a i r are f i n i t e sets. Every subset o f a f i n i t e
set i s f i n i t e .
I f a i s f i n i t e , then so i s the set composed o f a together with an additional
element. I n particular, the successor a+l o f a i s f i n i t e .

2 THEORY OF RELATIONS
Scheme of-injuction-for finite sets. If a condition f i s true for the emptyjet,
and if for every set a satisfying f and every set u , the set a u u j s m -
satisfies f , t+ i s true for every finite set.
If a set a and all i t s elements are finite, then the union u a i s finite.
This i s often expressed in the following form called pigeonhole principle: i f we
partition an infinite set into finitely many subsets, then a t least one of these
subsets i s infinite.
1.2. COUPLE OR ORDERED PAIR, CARTESIAN PRODUCT
Given two sets a, b the couple or ordered pair (a,b)
formed of the singleton { a ) and the (unordered) pair i a , b ) . This definition
goes back to KURATOWSKI 1921 (see also AJDUKIEWICZ). The set
the f i r s t term and b the second term of the couple. Clearly two couples are
equal i f f they have the same f i r s t and same second terms.
The Cartesian product a x b i s the set of couples (x,y) where x belongs
to a and y belongs to b .
FUNCTION, DOMAIN, RANGE
A function or mapping from a onto b i s a subset f of a x b such that
every element x of a appears as f i r s t term in exactly one couple (x.y)
belonging to f and every element y of b appears as a second term in a t
least one couple belonging t o f . The set a = Dom f i s called the domain,
the set b = Rng f i s the range of f .
For each element x of a , the second term y of the unique couple (x,y)
having f i r s t term x is denoted y = f(x) or y = fx and is called the
value of f on x , or the image of x under f .
For every superset c 7 Rng f we say that f i s a function from a into c .
THE TRANSFORMATION f" AN0 ITS INVERSE
If uc_ Dom f ,we denote by f"(u) the set of elements fx where x u .
The function thus denoted f" i s a function on the set of subsets of Dom f
and i s called the transformation associated with f . This transformation
preserves inclusion, in the sense that u c_ v implies f"(u) c f"(v) . However
strict inclusion i s not preserved.
If v C Rng f , then the inverse image of v by f , denoted (f- )"(v) , is
the set of elements x such that fx belongs to v . So we define the inverse
transformation associated w i t h f , denoted
inclusion as well as inclusion.
INJECTION, INVERSE FUNCTION, PERMUTATION, TRANSPOSITION
The function
implies fx # fx' for all x, x ' in Dom f . If a i s the domain, b the range,
i s the set {{a} , {a,b]
a i s said t o be
- -
1
(f-')" . I t preserves s t r i c t
f i s said to be an injection or injective function i f f x # x '

Chapter 1 3
then an i n j e c t i o n i s s a i d t o be a b i j e c t i o n from a onto b .
The inverse o f an i n j e c t i o n f i s denoted by f - l , so t h a t i n the case o f f
i n j e c t i v e , the transformation associated w i t h f - l coincides w i t h ( f - l ) " (the
l a t t e r e x i s t s f o r every f u n c t i o n
Given a f u n c t i o n f , i n j e c t i v e o r not, i f Oom f i s f i n i t e , then Rng f i s
f i n i t e . For f i n j e c t i v e , the converse i s true.
A permutation o f a i s a b i j e c t i o n from a onto a . Given two elements x, y
o f a , t h e transposition (x,y), i s the permutation o f a which interchanges
x and y and i s the i d e n t i t y on every other element o f a .
FIXED POINT LEMMA (KNASTER 1928, generalized by TARSKI 1955)
L e t a be a s e t and h a f u n c t i o n which takes each subset x o f a t o a subset
hx o f a . Suppose t h a t h i s increasing under inclusion: x c y implies t h a t
hx C- hy f o r every x, y 5 a . Then:
(1) there e x i s t sets x a majorized by h , i n t h e sense t h a t x 5 hx ; f o r
example x can be taken as t h e empty set;
(2) if x i s m a j o r i z e d A h , then hx i s majorized & h ;
(3) the union u o f a l l majorized subsets-satisfies hu = u .
f ).
1.3. RESTRICTION, EXTENSION, COMPOSITION
Given a function f w i t h domain a and a subset b o f a , we c a l l the r e s t r i c -
t& o f f t o b , denoted f!b , t h e s e t o f ordered p a i r s belonging t o f o f
which t h e f i r s t term belongs t o b .
P u t t i n g g = f / b , we say t h a t f i s an extension o f g t o the domain a .
We leave i t t o t h e reader t o define the composition g,f o f t h e functions
f and g , w i t h Dom(g,f) = Dom g n Rng f
EQUIPOTENCE, SUBPOTENCE
A s e t b i s s a i d t o be equipotent w i t h a
onto b .
A s e t b i s s a i d t o be subpotent w i t h a
potent w i t h b I A s e t b i s s t r i c t l y subp
w i t h a but a i s n o t subpotent w i t h b .
i f f there e x i s t s a b i j e c t i o n o f a
ff there e x i s t s a subset o f a equi-
k t w i t h a i f f b i s subpotent
By theorem 1.4 below, t h i s i s equi-
valent t o saying t h a t b i s subpotent but n o t equipotent w i t h a .
Every s e t equipotent w i t h a f i n i t e s e t i s i t s e l f f i n i t e . Every f i n i t e s e t i s
s t r i c t l y subpotent w i t h every i n f i n i t e set. Two f i n i t e sets are always comparable,
one being subpotent w i t h the other. I f a and b are f i n i t e , then the Cartesian
product a x b i s f i n i t e . If a i s f i n i t e , then so i s t h e power s e t p ( a ) .
A f i n i t e s e t i s not equipotent w i t h any o f i t s proper subsets. Equivalently, i f
a i s f i n i t e , then every i n j e c t i o n o f a i n t o a i s a permutation o f a .
0 Suppose t h a t f i s an i n j e c t i o n s a t i s f y i n g f"(a) c a . Take a subset m o f
a which i s minimal among a l l subsets x o f a s a t i s f y i n g f o ( x ) c x . Then

f"(f"(m)) c f"(m) by the i n j e c t i v i t y o f f : t h i s contradicts the minimality. 0
A set a i s said t o be Dedekind-finite i f f a i s not equipotent with any proper
subset o f i t s e l f (DEDEKIND 1888); i t i s Dedekind-infinite i n the opposite case.
Every f i n i t e set i s Dedekind-finite. The converse w i l l be proved i n 2.6 by using
the denumerable subset axiom (weaker than the axiom o f choice).
DEDEKIND-FINITE SET
1.4. BERNSTEIN-SCHRODER THEOREM
Given sets a and b ,if a i s subpotent with b and b subpotent with a ,
-then a i s equipotent with b . The following proof i s i n FRAENKEL 1953 and a t t r i -
buted t o WHITAKER. It does not use the notion o f integer, which i s used i n the
classical "mirror proof" ; see a1so SUPPES 1960.
0 Let f be an i n j e c t i o n from a i n t o b , and g be an i n j e c t i o n from b
i n t o a . It suffices t o f i n d a subset u o f a such that b-f"(u) i s sent
t o a-o by the function go , o r equivalently u = a - g"(b-f"(u)) . To do t h i s ,
consider the function which takes each subset x o f a i n t o a - g"(b-f"(x)) .
This function i s increasing under inclusion. By the f i x e d point lemma, the union
u o f a l l x such that x 5 a - g"(b-f"(x)) s a t i s f i e s the above.
1.5. CANTOR'S LEMMA
Let a be a set. There i s no function, i n j e c t i v e o r otherwise, with domain a
and range 9 (a) (set o f subsets o f a ).
CANTOR'S THEOREM
(1) Every set a i s s t r i c t l y subpotent with 9 (a) .
(2) I f a i s non-empty, then every set o f mutually d i s j o i n t subsets o f a
s t r i c t l y subpotent with 9 (a) .
1.6. EXPONENTIAL
Given sets a and b , the exponential o r power ab i s the set o f functions
from a
However aO = 0 f o r each non-empty set a . For each set a , the set 7 (a) o f
subsets o f a i s equipotent with a2 ,where 2 = { O , l ) ) .
We have the following equipotences. For b and c d i s j o i n t , (b"c)a i s equi-
potent with the Cartesian product (ba)x(ca) . The set '(a x b) i s equipotent
with the product (Ca)x(cb) . F i n a l l y C(ba) i s equipotent with (bxc)a .
i n t o b . Thus 'b = {O) = 1 f o r each b . I n p a r t i c u l a r '0 = 1 .
1.7. CHOICE SET AND CHOICE FUNCTION
Let a be a set o f non-empty mutually d i s j o i n t sets x . A choice set f o r a i s
a set whose intersection with each element x o f a i s a singleton.

Chapter 1 5
I f a i s f i n i t e , there i s always a choice set f o r a (proof by induction).
Let a be a set o f non-empty sets x . A choice function f o r a i s a function
which t o every element x o f a associates an element f x o f x . I f a i s
f i n i t e , then there i s a choice function f o r a .
AXIOM OF CHOICE (ZERMELO 1908)
Every set, even i n f i n i t e , o f non-empty mutually d i s j o i n t sets admits a choice set.
Equivalently every s e t o f non-empty sets admits a choice function.
An immediate consequence o f the axiom o f choice i s the following. Given a function
f , i n j e c t i v e o r otherwise, Rng f i s subpotent with Dom f . I n other words,
given a non-empty set a , every s e t o f mutually d i s j o i n t subsets o f a i s
subpotent with a .
Problem. Does the preceding statement imply the axiom o f choice (problem mentioned
i n RUBIN 1963 p. 5 note 1). A seemingly weaker consequence o f the axiom o f choice
i s the assertion that i s never s t r i c t l y subpotent with Rng f . This does
not follow from ZF alone i.e. from the axioms mentioned i n 5 1, 5 2 and 2.4; see
ch.10 exerc. 2, where a FRAENKEL-MOSTOWSKI model i s constructed with Dom f
s t r i c t l y subpotent with Rng f , a r e s u l t which i s transferable t o ZF via the
theorem o f JECH-SOCHOR (observation due t o HODGES).
.____
Dom f
1.8. GENERALIZED CARTESIAN PRODUCT
Let a
a i s the set o f choice functions which, t o each element ai o f a associate
an element o f ai . I f a reduces t o the p a i r b,c), we have again the Cartesian
product b x c o f 1.2. If a i s i n f i n i t e , i t follows from the axiom o f choice
that the Cartesian product o f a i s non-empty.
KONIG'S THEOREM
Let I be a non-empty set o f elements i (called indices), t o each o f which i s
associated a p a i r o f sets ai, bi with ai s t r i c t l y subpotent with bi .
Then the union o f the
product o f the bi (axiom o f choice i s used).
0 Suppose there exists a b i j e c t i o n h from u ai
the bi . For each i and each x o f ai , take the function hx E 17bi and
take i t s value ( h x ) ( i ) . Thus we define a function from ai i n t o bi . By the
axiom o f choice, the range o f t h i s function i s subpotent with ai , thus s t r i c t l y
subpotent with bi . Hence there i s an element ui o f bi which i s not the
value o f hx on i f o r any x i n ai . The choice function which t o each i
associates ui i s not i n h " ( u ai) : contradiction. We leave i t t o the reader
t o see that the union o f the i s subpotent with the product o f the bi . 0
be a non-empty set whose elements are non-empty. The Cartesian product o f
ai ( i 6 I ) i s s t r i c t l y subpotent with the Cartesian
onto Tr bi , the product o f
ai

Problem. Can the above theorem be proved from only the axioms of ZF in the case
where the set
Note that i f , in addition t o I being finite, we have for each index i that
T ( a i )
suffice for the proof.
For
asserts t h a t Dom f is never strictly subpotent with Rng f , or of ZF plus the
apparently weaker axiom which asserts t h a t if a (resp. a' ) is strictly
subpotent with b (resp. b' ) b, b ' disjoint, then a u a ' i s strictly
subpotent with b u b ' .
I of indices is finite w i t h cardinality greater than or equal to 2.
i s subpotent with bi , then by CANTOR'S lemma 1.5, the axioms o f ZF
Card I = 2 , KONIG's theorem is a consequence of ZF plus the axiom which
§ 2 - SECONDGROUP OF AXIOMS FOR ZF: FOUNDATION, I N F I N I T Y ,
SUBSTITUTION; ORDINAL, INTEGER, COUNTABLE SET
AXIOM OF FOUNDATION
The axiom of foundation i s the statement t h a t every non-empty set
element disjoint from a . I t follows that x $ x for any x . Moreover for any
XY Y i t i s impossible that x E y and Y Ex , etc.
The axiom of foundation was introduced by ZERMELO 1930, inspired by a statement
of von NEUMANN 1929. As t o i t s consistency, supposing t h a t all other axioms of
ZF are consistent, see exercise 1.
PREDECESSOR
Given a set a , the successor a+l = a v j a } i s distinct from a , since a $! a .
Moreover i f a+l = b+l then a = b ; otherwise we would have a € b+l with
a + b , so a E b and similarly b E a , contradicting the axiom of foundation.
Given a set c , the set whose successor is c (which is unique i f i t exists)
is called the predecessor of c , denoted by c-1 . Finally, given a set a and
i t s successor at1 , there i s no set x such t h a t a E x E a+l .
TRANSITIVE SET, TOTALLY ORDERED SET
A set a i s transitive i f f , for every x, y, conditions y 6 x E a imply y E a .
If a i s transitive and non-empty, then every element o f a i s a proper subset
of a . Also 0 E. a (0 i s the only element o f a which i s disjoint from a ).
Every union and intersection of transitive sets i s transitive.
If a i s transitive, then so i s a+l ,
A set a i s totally ordered (by membership relation) i f f , for every x, y of a ,
either X E . y or y E x or x = y . For example, all singletons are totally
ordered. However the singleton o f 1, i.e. 11) ={{O)): is not transitive.
The set {O,l,{l)} is transitive b u t not totally ordered. The set {0,(1))
neither transitive nor totally ordered. Every intersection of totally ordered sets
is totally ordered. A union of such sets is not necessarily totally ordered;
a admits an
i s

Chapter 1 I
however i f the s e t o f t o t a l l y ordered sets i s directed under i n c l u s i o n ( i . e . any
two such sets are included i n a t h i r d such set), then the union i s t o t a l l y ordered.
F i n a l l y i f a i s t o t a l l y ordered by E , then so i s a + 1 .
2.1. ORDINAL
An ordinal i s a t r a n s i t i v e s e t which i s t o t a l l y ordered by E . For example 0 ,
Every element o f an o r d i n a l i s an ordinal. The successor s e t o f an ordinal i s an
o r d i n a l . The predecessor ( i f i t e x i s t s ) o f an o r d i n a l i s an ordinal.
The i n t e r s e c t i o n o f any s e t o f ordinals i s an o r d i n a l .
An ordinal a i s s a i d t o be less than o r equal t o an o r d i n a l b , denoted a .4( b ,
i f f a € b o r a = b ; an o r d i n a l a i s s t r i c t l y less than b , denoted a < b ,
i f f a € b . Hence < i s synonymous w i t h 6 between ordinals. S i m i l a r l y a
(greater than o r equal t o ) and > ( s t r i c t l y greater than) are defined.
I f a,c b+l , then a s b o r a = b+l .
Given two ordinals a fi b , the condition a E b ( o r a < b ) i s equivalent
t o s t r i c t i n c l u s i o n a c b . Hence a & b i s equivalent t o a c_ b .
0 By t r a n s i t i v i t y a E b implies a c b .
Conversely, suppose t h a t a c b . L e t d E b-a be an element d i s j o i n t from b-a .
As d e b , t h i s d i s an o r d i n a l and d c b . Also d c_ a since d i s d i s j o i n t
from b-a . So e i t h e r d = a ( y i e l d i n g a € b ), o r d c a . I f the l a t t e r occurs,
l e t u E a-d so t h a t u ca c b . As b i s an o r d i n a l and u E b and d E b ,
we have e i t h e r u E d o r d E u o r u = d . If U E d , t h i s contradicts
u E a-d . If d E u , then since u E a ,we have d E a which contradicts
d e b-a . I f u = d , then d E a-d so d E a , again c o n t r a d i c t i n g d E b-a . 0
TRICHOTOMY
Given any two o r d i n a l s a, b, e i t h e r a € b b E a a = b .
0 As we know, the i n t e r s e c t i o n a n b i s an ordinal. E i t h e r a r b = a o r
a n b = b o r a n b i s s t r i c t l y included i n both a and b . I n the f i r s t
case a s b so a = b o r a c b and thus a € b . A s i m i l a r conclusion i s
reached i n the second case. I n t h e t h i r d case, we have a A b a and
a 0 b E b , so t h a t a n b belongs t o i t s e l f , c o n t r a d i c t i n g t h e axiom o f
foundation. 0
L e t a and b be two ordinals; i f b b a then b 3 a+l o r b = a .
We leave i t t o the reader t o define t h e maximum o r minimum ordinal o f a s e t o f
ordinals, denoted Max, Min . Every non-empty s e t a o f ordinals admits a mini-
-mum: take b belonging t o a and d i s j o i n t from a .
More generally we have t h e f o l l o w i n g scheme o f statements: given a condition ‘8
which i s s a t i s f i e d by a t l e a s t one ordinal, there i s a minimum ordinal s a t i s f y i n g f
Every t r a n s i t i v e s e t o f ordinals, every union o f a s e t o f ordinals i s an ordinal.
1 = t o } , 2 = {0,1) .

We leave i t t o the reader t o define upper bound and lower bound o f a set o f ordi-
nals. Given a set u o f ordinals, we denote the union o f u by Sup u . It i s
the supremum o f u , i . e . the least upper bound o f u .
I f o( i s an ordinal and u a set o f ordinals such that /3 e u implies f i s ,
then Sup u 6 o( . I n other words i f a < Sup u , then there exists an ordinal
i n u with (3 > q .
2.2. ORDINAL-INDEXED SEQUENCE, o( -SEQUENCE; EXTRACTED SEQUENCE
Given an ordinal o( , an d - z e q u e x e , or ordinal-indexed sequence, i s a function
with domain o( . I n t h i s case oc i s the length o f the sequence.
Given a sequence u , the elements o r =of u are a l l ordered pairs (i,ui)
f o r which the f i r s t term i i s an ordinal s t r i c t l y less than OC . The i ' s are
called indices o f u , o r u i s indexed by i < o( . The second terms o f the
ordered pairs (which are a r b i t r a r y sets) are called the values o f u and denoted
ui o r u ( i ) .
I n the particular case o f an h -sequence with ordinal values, we leave i t t o the
reader t o define increasing, decreasing, s t r i c t l y increasing and s t r i c t l y decrea-
sing sequences.
Given an ordinal o( and an 4 -sequence u ,we define an extracted sequence
from u t o be a sequence with length 1364 , obtained by composition v o f
u with h , where h i s a s t r i c t l y increasing &-sequence with values i n O( ;
so v = u,h and v. = u f o r each i < f j . The notion o f extracted sequence
i s reflexive and transitive, but not antisynnnetric. For instance, by the axiom o f
i n f i n i t y introduced i n 2.4 below, given two d i s t i n c t sets a, b, we w i l l define
the a-sequences a,b,a,b,.. and b,a,b,a,.. ,each extracted from the other.
i h ( i )
2.3. INTEGER, n-ELEMENT SET, WORD, n-TUPLE
By non-negative integer, o r integer, o r natural number, we mean a f i n i t e ordinal.
Every element o f an integer i s an integer. Every non-zero (i.e. non-empty) integer
has an integer predecessor.
I f a i s an integer and b E a ( o r b < a ) , then b i s s t r i c t l y included i n a .
As a i s f i n i t e , b i s s t r i c t l y subpotent with a . Thus equipotent integers
are identical.
We thus have the scheme o f induction: i f a condition f holds f o r 0 , and i f f o r
each integer a the condition f ( a ) implies f?(a+l) , z/fholds f o r every
integer.
Every f i n i t e set i s equipotent with an integer. Given a set a , t h i s integer i s
called the cardinal, o r c a r d i n a l i t y o f a and denoted Card a .
A set equipotent with an integer n i s called an n-element set.
A f i n i t e sequence o r word i s an n-sequence, where n
of length n i s called an n-t=.
i s an integer; such a word-

Chapter 1 9
When r e s t r i c t e d t o words, the notion o f extracted sequence becomes antisymmetric;
i.e. two words each o f which i s extracted from the other are identical.
2.4. AXIOM OF INFINITY, SUCCESSOR AND LIMIT ORDINAL
The axiom o f i n f i n i t y asserts the existence o f an i n f i n i t e set. A more useful
and stronger version asserts the existence o f a Dedekind-infinite set. More
precisely the existence o f a set a containing the element 0 and such t h a t
-i f x belongs t o a , then the successor x+l = x u ( x ) belongs t o a . With
an appropriate application o f the separation scheme, the axiom o f i n f i n i t y
yields the existence o f the set o f integers, denoted by 0 . The set w i s an
i n f i n i t e ordinal, the smallest ordinal > 0 without a predecessor.
A limit ordinal i s an ordinal without a predecessor. A successor ordinal i s an
ordinal with a predecessor.
SUBSTITUTION SCHEME
A t t h i s point we replace the separation scheme by the more general substitution
scheme (due t o FRAENKEL 1925), o f which the reader i s assumed t o be familiar.
With t h i s scheme we can define, f o r example, L3+ W = w . 2 : beginning with the
set w o f integers, associate t o each integer i the ordinal w+i defined
below i n section 3.1. Then using the substitution scheme define the set o f ~ + i
as i runs through . Another example: denote by No the set G) o f integers,
and f o r each integer i
substitution scheme allows one t o define the set o f Ni f o r i belonging t o w .
The axioms previously introduced, from 5 1 t o the present 5 2.4 (not including
the axiom of choice nor i t s weakened versions such as choice among f i n i t e sets),
are called the axioms of ZF. If no special assumption i s e x p l i c i t e l y mentioned
i n a theorem, then t h i s indicates t h a t the theorem i s proved i n ZF alone.
If, however, the axiom o f choice o r other supplementary axioms ( f o r the most
p a r t weakened versions o f the axiom o f choice, stated below) are used, then we
clearly indicate such. We have already done t h i s f o r KONIG's theorem i n 1.8.
Recall t h a t the axiom of choice has been proved consistent with ZF ( i f ZF i t s e l f
i s consistent) by GODEL 1938. The negation o f the axiom o f choice has been
proved equiconsistent with ZF by COHEN 1963 (see the Bibliography COHEN 1966).
l e t Ni+l = y ( N i ) (the set o f subsets o f Ni) ; the
2.5. DENUMERABLE SET, COUNTABLE SET, COUNTABLE AXIOM OF CHOICE
A set i s said t o be denumerable, resp. countable, i f i t i s equipotent, resp.
subpotent with w , the set o f integers. ZF alone suffices t o show t h a t the
union o f two denumerable sets, the Cartesian product o f two denumerable sets,
and the set o f a l l f i n i t e subsets o f w are a l l denumerable.
Following 2.2, we c a l l an &-sequence a sequence o f length w , hence indexed
by the set o f integers.

The countable axiom o f choice i s a p a r t i c u l a r case o f the axiom o f choice. It states
t h a t f o r every countable s e t o f non-empty d i s j o i n t sets, there i s a choice set.
This axiom i s s t r i c t l y weaker than the axiom o f choice; i . e . i f ZF i s consistent,
then there i s a model o f ZF and countable choice which s a t i s f i e s the negation of
the general axiom o f choice (JECH 1973).
Note t h a t t h e countable axiom o f choice implies t h a t every denumerable union o f
denumerable sets i s denumerable. Indeed, t h i s axiom allows one t o choose, f o r each
o f the denumerable sets i n t h e union, a b i j e c t i o n from t h a t s e t onto t h e integers.
On the other hand, the above statement i s n o t provable from ZF alone: there i s a
model o f ZF itn which t h e continuum i s a denumerable union o f denumerable sets
( A z r i e l LEVY, unpublished).
2.6. DENUMERABLE SUBSET AXIOM
This axiom states t h a t every i n f i n i t e s e t has a denumerable subset. It follows
from the countable axiom o f choice.
0 L e t a be an i n f i n i t e s e t . For each i n t e g e r i , associate the s e t o f i - t u p l e s
o f elements from
one o f these i-tuples. It remains t o take t h e W-sequence formed from the terms
o f the chosen 1-tuple, 2-tuple, ... . 0
The denumerable subset axiom i s s t r i c t l y weaker than t h e countable choice (JECH 1973).
Let a
(1) there e x i s t s a denumerable subset o f a ;
(2) there e x i s t s a b i j e c t i o n o f a
a i s Dedekind-infinite (see 1.3);
(3) there e x i s t s a choice f u n c t i o n f which t o each f i n i t e subset x of a
associates an element f x i n t h e complement a-x .
Consequently, thedenumerable subset axiom i s equivalent t o saying t h a t f i n i t e n e s s
coincides w i t h Dedekind-finiteness. However w i t h ZF alone, there can e x i s t an
i n f i n i t e s e t having f o r each i n t e g e r i a subset equipotent w i t h i , y e t having
no denumerable subset.
a . By countable choice, we can associate t o each i n t e g e r i
be an i n f i n i t e set; t h e f o l l o w i n g three conditions are equivalent:
onto a proper subset o f a ; i n other words,
2.7. Having defined the integers, we can now complete the i n i t i a l remarks from
5 2 by adding that, w i t h the axiom o f foundation, there are no O-sequences u
w i t h ui+l belonging t o ui f o r each i n t e g e r i . I n p a r t i c u l a r , f o r every i n t e -
ger r , there i s no cycle u1C u2 E ...E ur E u1 .
2.8. The axiom o f foundation i s equivalent t o t h e f o l l o w i n g axiom scheme.
Let f be a condition which holds f o r 0 and such t h a t , i f holds f o r each ele-
ment o f a given s e t a , then f holds f o r a . Under these hypotheses, f holds
f o r every set. Note t h a t we can eliminate the hypothesis " f holds f o r 0 ";t h i s
-

Chapter 1 11
being a p a r t i c u l a r case o f the second hypothesis, made precise as follows: e i t h e r
there e x i s t s an element o f a s a t i s f y i n g "not f I' , o r a s a t i s f i e s f .
0 L e t a be a s e t which f a l s i f i e s t h e axiom o f foundation, and l e t '6 be the
condition holding f o r 0 and f o r every s e t which does n o t belong t o a . Then
(e s a t i s f i e s our hypotheses, b u t 'if does n o t h o l d f o r every element o f a .
Conversely, l e t be a c o n d i t i o n s a t i s f y i n g our hypotheses, b u t such t h a t the s e t
a does n o t s a t i s f y . L e t al be the s e t o f elements o f a s a t i s f y i n g "not f".
Let a2 be t h e s e t o f elements o f t h e union u al s a t i s f y i n g "not If: " . L e t a3
be t h e s e t o f elements o f u a2 s a t i s f y i n g "not I$", etc. Then t h e union o f the
ai (ii n t e g e r ) f a l s i f i e s the axiom o f foundation. 0
2.9. A necessary and s u f f i c i e n t condition f o r a s e t
f o r every f i n i t e sequence ao, al, ... , an w i t h a. = a and ai+l belonging t o
ai f o r i< n , every ai i s t r a n s i t i v e . I n other words, an o r d i n a l i s a s e t
which i s h e r e d i t a r i l y t r a n s i t i v e ; use 2.1: every t r a n s i t i v e s e t o f ordinals i s
an ordinal.
Equivalently, a i s an o r d i n a l i f f a and a l l elementsof a are t r a n s i t i v e
(see f o r instance POWELL 1975 p. 223).
Analogously, we leave i t t o the reader t o prove t h a t a s e t a i s an i n t e g e r
i f f a i s empty o r a i s a successor set, and every element o f a i s e i t h e r
empty o r a successor s e t (communicated by HATCHER 1977).
a t o be an ordinal i s t h a t
2.10. AXIOM OF CHOICE FOR FINITE SETS
Now t h a t i n f i n i t e sets have been introduced, we i n d i c a t e here an important
weakening o f the axiom o f choice, which asserts the existence o f a choice s e t
f o r every s e t o f non-empty, f i n i t e , mutually d i s j o i n t sets. This weakened form
i s n o t implied by and does n o t imply t h e countable axiom o f choice from 0 2.5,
nor the denumerable subset axiom o f 2.6. I n f a c t the axiom o f dependent choice,
which i s stronger than countable choice, does n o t imply choice among f i n i t e
sets: see ch.2 5 1.6.
2.11. We s h a l l c a l l induction, o r t r a n s f i n i t e induction, the f o l l o w i n g reasoning.
Suppose t h a t i f a condition 'f holds f o r every o r d i n a l s t r i c t l y less than o(
then If holds f o r 4 ; under t h i s hypothesis, f holds f o r every o r d i n a l . This
i s a form o f the scheme s t a t e d i n 5 2.1: i f "not "
i s s a t i s f i e d by a t l e a s t one
ordinal, there i s a l e a s t ordinal s a t i s f y i n g "not e''. Often, induction i s broken
up i n t o a p r o o f f o r 0, a p r o o f f o r the t r a n s i t i o n between an a r b i t r a r y o r d i n a l o(
and i t s successor o( + 1 , and a proof f o r o(
A d e f i n i t i o n by recursion i s made by introducing a statement f ( o ( ,a) which
uniquely associates a s e t a t o each o r d i n a l o( . This statement w i l l usually be
a l i m i t o r d i n a l .

o f the following form. "There exists one and only one function f with domain
d + 1 (the successor o f o(),such t h a t the i n i t i a l ordered p a i r
f (where u i s a r b i t r a r i l y given), the f i n a l ordered p a i r (o( ,a) belongs t o
f , and such t h a t f o r each (3 o( the p a i r ( (3 ,b) belongs t o f , provided
that b has been obtained i n a certain (suitably defined) manner from the set o f
ordered pairs belonging t o f with f i r s t term < 13 ' I . Because o f the uniqueness
o f f , when q'>d, , the function f ' corresponding t o oc' w i l l be an exten-
sion o f f t o the domain a( '+1 . Some examples o f d e f i n i t i o n by recursion:
sum, product, exponentiation f o r ordinals i n 0 3; aleph rank i n 0 6.4. The defini-
t i o n o f fundamental rank i n 0 5.2 i s also by recursion, i f one begins by associa-
t i n g t o each ordinal o( the set o f a l l sets with fundamental rank o( .
Note t h a t d e f i n i t i o n by recursion using the axioms o f ZF i s easier t o j u s t i f y
than d e f i n i t i o n by simple recursion i n f i r s t - o r d e r Peano arithmetic, such as i s
generally presented today (however, the original t e x t o f PEANO 1894 i s written
i n second-order l o g i c ) . I n order t o j u s t i f y d e f i n i t i o n by recursion i n f i r s t -
order arithmetic, one i s led, i n the manner o f GOOEL 1931, t o use the "Chinese
remainder theorem". For instance, one defines b = a! as an abbreviation f o r
the following: " there e x i s t two integers u, v such t h a t the remainder a f t e r
division o f u by v+l i s 1 , the remainder a f t e r division o f u by
(a+l)v + 1 i s b , and for each i ( 1 s i d a) one obtains the remainder a f t e r
division o f u by ( i + l ) v + 1 from the remainder a f t e r d i v i s i o n o f u by
i v + 1 by multiplying the l a t t e r by i+l" .
(0,u) belongs t o
5 3 - REVIEW OF ORDINAL ALGEBRA, CANTOR NORMAL FORM, INDECOMPOSABLE
ORDINAL
3.1. SUM
We say that o( +/3 = Y (where d,b,8 are ordinals) i f f t h e r e exists a function
f with domain fi +1 (hence, f o r each u& fi there i s one and only one ordered
p a i r belonging t o f with f i r s t term u ), such that the i n i t i a l p a i r (0, m )
and f i n a l p a i r ( / 5 , ) belong t o f ; i f (u,v) belongs t o f where u < fs ,
then (u+l,v+l) belongs t o f ; and f i n a l l y such t h a t i f f contains as elements
ordered pairs (x,y) f o r which the f i r s t terms x admit a supremum Sup x 4 (5,
then the p a i r (Sup x, Sup y ) belongs t o f .
Given o( and /3 , the reader can prove by induction on /3 the existence and
uniqueness o f the preceding function, hence the existence and uniqueness o f the
ordinal 5 = A + fi . I n the same manner, one proves f o r every o( ,/s the equa-
l i t i e s d + O = O+o( = % and oC+( /5+1) = ( *+f3)+1 , and f o r every o( and
every set o f ordinals u the supremum equality o( +(Sup u) = Sup(& + u) .
For a l l o( ,13P 0 we have o(+/3, >o( . For a11 o( ,p we have 4+ >,(3 where

Chapter 1 13
equality i s possible with non-zero o( : f o r instance 1 + 4) = G, .
The supremum equality does not hold on the l e f t : i f i i s an arbitrary integer
then S u p ( i + w ) = ~d # (Sup i ) + LJ = ~ d +t.3.
Ordinal addition i s associative. Commutativity holds f o r integers, o r f i n i t e
ordinals; however 1 + CJ = c d # id+ 1 .
For a l l O( and f i 2 o( , there exists one and only one satisfying
CA + 8 = 0 ; t h i s 2( i s called the difference /3 - o( .
The inequality 6 implies o(+/3$ o< + &' and conversely. Also the
same r e s u l t f o r s t r i c t inequality < . Hence addition i s l e f t cancellable, i.e.
The inequality % $ implies o( + 6 + I f . This does not hold i n
general f o r < , as 0 + W = 1 + W . Hence addition i s not r i g h t cancellable.
F i n a l l y the ordinal 1 and consequently every f i n i t e ordinal i s absorbed by
every i n f i n i t e ordinal, i n the sense t h a t 1 + o( = o( f o r q i n f i n i t e .
w + /3 = O( + II implies b =8 .
3.2. PRODUCT
We say t h a t o( . /s = ?f i f f there exists a function f w i t h domain fi + 1 ,
such that the i n i t i a l p a i r (0,O) and the f i n a l p a i r ( 0 ,g ) belong t o f ,
and such t h a t i f (u,v) f where u < fs then (u+l,v+o( ) 6 f , and such
that i f (x,y) E f f o r a l l x belonging t o a set which admits a supremum
Sup x ,< fi then (Sup x , Sup y ) E f .
For a l l o( ,p we have o( .O = 0 . N = 0
For every o( and every set o f ordinals u , we have the supremum equality
d .(sup u) = sup(@ .u) . Moreover d.0= O i s equivalent t o o ( = O o r /3= O .
Ordinals o f the form o(.u , with o( f i x e d and u an a r b i t r a r y ordinal, are
called the multiples o f o( . For example 0 i s a multiple o f every ordinal.
Every multiple o f M I augmented by o( ,y i e l d s a multiple o f o( . The supremum
o f a set o f multiples o f d i s a multiple o f o( . F i n a l l y every multiple o f o(
i s obtained from 0 by these two indicated orocesses. More rigorously i f a condi-
t i o n i s true f o r 0 and i s preserved i n the passage from an ordinal u t o u+w
as well as i n the passage t o supremum, then t h i s condition i s true f o r every
multiple o f o( .
The supremum equality on the r i g h t , given above, does not hold on the l e f t : i f
i designates an a r b i t r a r y integer, then Sup(i.2) = W # (Sup i ) . 2 = CJ .2 .
Multiplication i s associative and d i s t r i b u t i v e on the r i g h t : r . ( M + / 3 ) =r.q+r,(,pI
D i s t r i b u t i v i t y on the l e f t and commutativity hold f o r integers; however
( w + l ) . w = c J . o # w . b + 1 . ~ 3 and 2 . ~ = ~ # ~ 3 . 2 .I t c a n h a p p e n t h a t c ( . p
i s not a multiple of (3 , e.g. (0+1).2 = w . 2 + 1 : i t i s not a multiple o f 2 .
For 4 # 0 the inequality 1)4 $ implies & . (3 6 &.$ and conversely. The
same r e s u l t holds f o r s t r i c t inequality. Thus multiplication i s cancellable on
and d .( &+1) = o(.0 + o( .

the l e f t except f o r 0 ; i . e . f o r O( non-zero g.fs = q . v implies 0= d .
The inequality o( & f s implies o( .y,r 0 . 8
since 1.a= 2 . w = W . M u l t i p l i c a t i o n i s thus not cancellable on the r i g h t .
Given two ordinals d and f i #0 , there i s a unique ordinal called the
quotient, and a unique ordinal E called the remainder i n the d i v i s i o n o f o(
by fi , w i t h g = f i $ + and E <' 0: consequence o f the existence o f a
maximum ordinal u such t h a t /s u 4 4 .
. This does not subsist f o r <
3.3. POWER OR EXPONENTIATION
We say t h a t &Is =
t h a t the i n i t i a l p a i r (0,l) and the f i n a l p a i r ((3,&) belong t o f , and
such t h a t i f (u,v)E f where u < (3 then ( u t l . v . 4 ) f , and such t h a t i f
(x,y) E f f o r x belonging t o a set which admits a supremum Sup x 6 (3, then
For a l l cx,p we have oCo = 1 and cc('+'I= 4'. 4 . For a l l o( and every set
o f ordinals u ,we have the supremum equality o( (sup
Moreover W p = 0 i s equivalent w i t h o( = 0 and f i # 0 . The equality M p = 1
i s equivalent t o o( = 1
Ordinals o f the form o( , with cx f i x e d and u an a r b i t r a r y ordinal, are called
powers o f O( . For example 1 i s a power o f every r% . If v i s a power o f o(
then so i s v.* . The supremum o f a set o f powers o f i s a power o f 4 .
Finally, every power o f o( i s obtained from 1 by these two processes.
The supremum equality on the r i g h t given above does not subsist on the l e f t : i f
i designates an a r b i t r a r y integer, then Sup(i ) = W # (Sup i)'= W
We have o( ( * + '
We have ( ~ & f i ) ~ =&(b'r)for a l l o<,p,8 . It i s impossible i n general t o
interchange terms o f the product i n the exponent:
The equality (ac).(bc) = (a.b)c which holds f o r integers, does not subsist i n
general, even f o r f i n i t e exponents: = o .2 .
This equality does not subsist f o r a, b f i n i t e and an i n f i n i t e exponent:
( 2 L J ) . ( Z W ) = L 3 2 # 4 W = w .
For o( >/ 2 , the inequality f34 21 implies NP< c ( ~and conversely; same
r e s u l t with < . Thus we have cancellation:
The inequality o( 4 -fir ; t h i s does not subsist f o r <
since z W = 3 w = w .
F i n a l l y f o r o(( f s the ordinal w4 i s absorbed by &*, i . e . aa+W f i = k)?
i f f there e x i s t s a function f with domain fi +1 , such
(SUP X I SUP Y) E f .
= Sup (oi ') .
o r /3 = 0
2 2
.
= &fi.4r for a l l O( ,P, r . It i s impossible i n general
t o interchange terms i n the product; f o r example 2( w+l)=w.2 f 2 1 . 2 w = w .
(2 ")' = W2 # 2('*@) =
2 2
( Q ~ ) . Z ~= u 2 . 4 # ( w . 2 )
= o( 'implies fs = b' .
3.4. Given two ordinals o( and /3 >/ 2 , there i s a unique ordinal which i s the
maximum exponent among the ordinals u s a t i s f y i n g fi < 4 .

Chapter 1 15
3-Given o( and (3 2 2 and the maximuA exponent 8 such t h a t /3 6 o( , there
e x i s t s a maximum o r d i n a l s such t h a t ( f i r ). $,< o( . Moreover
Given b ,
2 < (-5
and a s t r i c t l y decreasing, thus f i n i t e sequence o f ordinals
8 2 $ ( I ) '> g(2), ... and a corresponding se uence o f ordinals t ( l ) , $(2),
... each s t r i c t l y less than /3 , we have bx>fi9(').$(1) t fig(*).$(2) t ...
(proof by induction on 8 ) .
3.5. CANTOR NORMAL FORM
Given 4 and f3 >, 2 , there e x i s t s a decomposition o f o( i n t o a f i n i t e sum o f
-terms f i r , 6 , w i t h c o e f f i c i e n t s $< /3 and exponents s t r i c t l y decreasing.
Furthermore t h i s decomposition i s unique. It i s c a l l e d the Cantor decomposition
o f o( i n t o powers o f 0 o r Cantor normal form o f 4 i n base f , . I n t h e case
t h a t fi = u , the c o e f f i c i e n t s 6are integers.
3.6. DECOMPOSABLE AND INDECOMPOSABLE ORDINAL
An o r d i n a l o( i s c a l l e d decomposable i f f there e x i s t & < q and r'c.Cwith
then every sum o f two non-zero o r d i n a l s which i s equal t o o( has second term
equal t o d , and conversely.
A non-zero o r d i n a l o( i s indecomposable i f f O( i s a power o f a.This follows
from the existence and uniqueness o f the Cantor decomposition i n t o powers o f LC: ,
together w i t h the absorption statement (end o f 3.3).
o( = fl+8 ; otherwise o( i s c a l l e d indecomposable. I f o( i s indecomposable,
5 4 - EQUIPOTENT WITH THE CONTINUUM, CONTINUUM HYPOTHESIS, REAL
4.1. EQUIPOTENT WITH THE CONTINUUM
A s e t i s s a i d t o be equipotent w i t h t h e continuum i f f i t i s equipotent w i t h ?(a),
the power s e t o f the integers, o r equivalently w i t h "2 , t h e s e t o f functions
on w t a k i n g values 0 o r 1 . By CANTOR'S theorem 1.5, every countable s e t i s
s t r i c t l y subpotent w i t h 9( IC) ) .
Let a , b be two d i s j o i n t denumerable sets. By 1.6 we have t h a t a2 x b2 i s
equipotent w i t h (a " b)2 . Hence t h e Cartesian product o f two sets each equi-
potent w i t h t h e continuum i s i t s e l f equipotent w i t h the continuum. The same
r e s u l t holds f o r the Cartesian product o f a countable s e t w i t h a s e t which i s
equipotent w i t h the continuum.
S i m i l a r l y 2) i s equipotent w i t h ( w w)2 . Hence a i s a s e t
equipotent w i t h the continuum, then t h e s e t o f &-sequences w i t h values i n
i s a l s o equipotent with the continuum.
4.2. I f we subtract an a r b i t r a r y denumerable subset a from a s e t c equipotent
with the continuum, then the d i f f e r e n c e c-a i s equipotent w i t h the continuum.
a

This i s a special case o f the following proposition.
-Let a be an i n f i n i t e set which i s equipotent with the Cartesian product 2xa ,
and l e t c = y ( a ) . Then the difference set, obtained by removinq from c an
a r b i t r a r y subset which i s e q u i p o t e n t w i g a , i s equipotent with c .
0 Since a i s equipotent with 2xa , the set c , which i s equipotent with a2 ,
i s also equipotent with cxc by 1.6. Hence the difference o f c and a subset
which i s equipotent with a i s equipotent with the difference o f cxc and the
range o f a b i j e c t i o n f on a . Each element x o f a i s associated t o an
ordered p a i r f x = (y,z) o f elements y, z o f c . Let us associate t o each x
the f i r s t term y o f t h i s p a i r . The function thus obtained has domain a and
cannot have range c = ?(a) , by CANTOR'S lemma 1.5. Thus there exists an element
u o f c f o r which (u,z) i s not the value by f o f an element o f a , f o r any
z belonging t o c . Hence the difference o f cxc and f " ( a ) includes a subset
which i s equipotent with
with c . 0
c , and so by BERNSTEIN-SCHRODER 1.4 i s equipotent
4.3. Let a be a set equipotent with the continuum. For every p a r t i t i o n o f a
i n t o denumerably many subsets, one o f the subsets i s equipotent with the
continuum (uses the axiom of choice).
0 Suppose on the contrary t h a t there i s a p a r t i t i o n o f
ai ( i integer), and t h a t every
theorem 1.8 (axiom o f choice), the union a o f the ai i s s t r i c t l y subpotent
with the Cartesian product o f an a-sequence o f sets, each equipotent with the
continuum. But t h i s Cartesian product i s equipotent with the continuum: contra-
diction. 0
a i n t o d i s j o i n t subsets
ai
i s s t r i c t l y subpotent with a . Then by KONIG's
4.4. CONTINUUM HYPOTHESIS, GENERALIZED CONTINUUM HYPOTHESIS
The axiom called continuum hypothesis asserts the non-existence o f a set which
i s s t r i c t l y intermediate, with respect t o subpotence, between o and y(a ) .
This axiom i s l o g i c a l l y independent o f ZF, and even o f ZF plus the axiom o f
choice (COHEN 1963, see Bibliography 1966).
The axiom called generalized continuum hypothesis asserts the non-existence o f
a set s t r i c t l y intermediate, with respect t o subpotence, between a and p ( a ) ,
f o r every i n f i n i t e set a . When added t o the axioms o f ZF, t h i s implies the
axiom o f choice (see ch.2 exerc. 1).
4.5. REAL
We leave i t t o the reader t o redefine positive and negative integer, and then
real, as an ordered p a i r formed from an integer which i s called the integer
part, and an i n f i n i t e set o f non-negative integers. The l a t t e r set w i l l be
i d e n t i f i e d with an W-sequence o f terms ui ( i non-negative integer) with
-
-

Chapter 1 17
ui = 0 o r 1 according t o whether i belongs t o the i n f i n i t e set o f integers
or not. This sequence i s called the binary expansion o f the real, which always
contains i n f i n i t e l y many occurrences o f zero.
The notions o f rational real and dyadic real, i.e. rational whose denominator i s
a power o f 2 , are assumed t o be familiar, as well as the denumerability o f the
set o f rationals.
The set o f reals i s equipotent with the continuum: remove from the set o f a l l sets
o f integers, the denumerable set o f f i n i t e sets o f integers, and use 4.2.
We leave i t t o the reader t o define the ordering on the reals: less than o r equal
t o ( 3 ) , and the related s t r i c t inequalities.
Also the reader can define the notions o f dense, cofinal, c o i n i t i a l set o f reals
(an example being the rationals o r the dyadic reals). The reader can define a
closed, open, half-open i n t e r v a l o f reals, an i n i t i a l , f i n a l interval, an _upper
-bound and lower bound o f a set o f reals, the maximum, the minimum, a real valued
sequence which i s s t r i c t l y ( o r otherwise) increasing, decreasing.
Every set o f mutually d i s j o i n t intervals o f reals which are not reduced t o
singletons i s countable: enumerate the rationals and associate t o each interval
the f i r s t rational which belongs t o it.
Consequently, every s t r i c t l y increasing ( o r s t r i c t l y decreasing) ordinal-indexed
sequence o f reals i s countable.
( 3) , greater than o r equal t o
4.6. DEDEKIND'S THEOREM
I f we p a r t i t i o n the reals i n t o an i n i t i a l interval a and i t s complement the
f i n a l interval b , both non-empty, then e i t h e r a has a maximum element or
b has a minimum element.
Consequently, f o r any s e t a o f reals, i f there exists an upper bound, then
there exists a least upper bound called the supremum o f a and denoted Sup a .
Analogous d e f i n i t i o n o f the infimum which i s denoted I n f a . I n other words,
f o r every set a o f reals, there exists a smallest i n t e r v a l (with respect t o
inclusion) including a : the interval ( I n f a , Sup a) which i s closed, open
or half-open, i n i t i a l , f i n a l o r containing a l l the reals, depending on the case.
When useful, we w i l l use the
presumed t o know.
and product o f reals, which the reader i s
4.7. To see some i n i t i a l d i f f i c u l t i e s provided by the axiom o f choice, which
indicate t h a t t h i s axiom i s not "obvious", note that i t i s impossible i n ZF plus
the axiom o f choice, t o define and prove uniqueness o f a choice function which
associates t o each non-empty set o f reals one o f i t s elements. Similarly i t i s
impossible t o uniquely define a choice set picking one function from each p a i r
of real functions h,-h ,where f o r each real x , the value o f -h

on x i s the additive inverse o f h(x). To obtain a proof o f uniqueness, com-
pleting the existence (which i s guaranteed by the axiom o f choice), i t i s necessary
f o r example t o add t o ZF the axiom o f c o n s t r u c t i b i l i t y o f GODEL 1940.
§ 5 - TRANSITIVECLOSURE, HEREDITARILY FINITE SET, FUNDAMENTAL RANK,
CARDINAL
5.1. TRANSITIVE CLOSURE
For every set a , there e x i s t t r a n s i t i v e supersets o f a , and among these there
exists one which i s included i n a l l the others. This set i s formed from the values
o f a l l f i n i t e sequences xl, ...,xh (h integer) such t h a t x1 € a and xi+l E xi
f o r each i (1 6 i < h) . We shall c a l l t h i s set the t r a n s i t i v e closure of a .
For each non-empty s e t a ,the t r a n s i t i v e closure o f a i s the union of a
together w i t h the t r a n s i t i v e closures o f the elements o f a .
I f a s b then (closure o f a ) 5 (closure o f b ) ,
HEREDITARILY FINITE SET
A h e r e d i t a r i l y f i n i t e set i s a set whose t r a n s i t i v e closure i s f i n i t e . For instan-
ce, every f i n i t e t r a n s i t i v e set i s h e r e d i t a r i l y f i n i t e . I n p a r t i c u l a r every inte-
ger (i.e. every f i n i t e ordinal) i s h e r e d i t a r i l y f i n i t e .
The singleton o f 1 i s non-transitive y e t h e r e d i t a r i l y f i n i t e .
Every h e r e d i t a r i l y f i n i t e set i s f i n i t e , as it i s included i n i t s t r a n s i t i v e
closure which i s f i n i t e . Every element and every subset o f a h e r e d i t a r i l y f i n i t e
set i s h e r e d i t a r i l y f i n i t e . Every f i n i t e set o f h e r e d i t a r i l y f i n i t e sets i s here-
d i t a r i l y f i n i t e . Similarly f o r f i n i t e unions, f i n i t e Cartesian products, and the
power set o f h e r e d i t a r i l y f i n i t e sets.
A necessary and s u f f i c i e n t condition f o r a set a t o be h e r e d i t a r i l y f i n i t e i s
that, f o r every f i n i t e sequence xO, ...,xh ( h integer) with xo = a and
xi+l E xi for each i< h , the terms xi are f i n i t e .
-
5.2. FUNDAMENTAL RANK
Let a be a set and c be the t r a n s i t i v e closure o f the singleton ia 1 . We say
t h a t the ordinal o( i s the fundamental rank o f a , i f there exists a function
f with domain c , taking ordinal values 4 o( , such t h a t the i n i t i a l ordered
p a i r (0,O) and the f i n a l ordered p a i r (a, cx ) belong t o f : so t h a t f ( 0 ) = 0
and f(a) = d ; and such t h a t i f u E c then the value f ( u ) i s the smallest
ordinal s t r i c t l y greater than f ( x ) f o r a l l x belonging t o u .
It follows from the axiom o f foundation t h a t every set has a unique fundamental
rank. Indeed, the empty set 0 has rank 0 . Suppose t h a t a i s non-empty and
t h a t every element o f a has a rank. Then by the preceding definition, a has
rank equal t o the smallest ordinal which i s s t r i c t l y greater than the ranks of
-

Chapter 1 19
all its elements. The existence of rank results from the axiom of foundation i n
the form of scheme 2.8.
For every ordinal oc the fundamental rank i s g .
5.3. For every ordinal o( , there is a s e t
t t . 4 . Moreover V, has fundamental rank o( .
0 Obvious for 0 since Vo is empty. If this is true for cl( , then i t i s true
f o r @ + l with Vq +1 = s e t of elements and subsets of V,. Finally for q
a limit ordinal, V, is the union of the Vi for i s t r i c t l y less than o( . 0
Note that for each ordinal & , the s e t Vatl - V, of sets of rank o( i s non-
empty, since V, and 4 belong t o this set. For i an integer, or f i n i t e
ordinal, the s e t of sets of rank i is f i n i t e . I t follows that every infinite
s e t has rank a t least equal t o w .
Note that a s e t i s hereditarily f i n i t e i f f its fundamental rank is f i n i t e .
The set of hereditarily f i n i t e sets is the intersection of a l l sets which contain
0 and which, i f they contain x and y , also contain x u { y ) as an element.
Vd o f a l l sets of ranks s t r i c t l y less
-
5.4. CARDINAL, OR CARDINALITY
Given a set a , consider sets equipotent w i t h a and among these, those of mini-
mum fundamental rank. By the preceding, these form a non-empty s e t which we call
the cardinal or cardinality of a , denoted by Card a : definition from SCOTT 1955.
Thus every set has a cardinal, and two sets are equipotent i f f they have the same
cardinal. Note that every s e t a i s equipotent, not t o Card a , but to an arbi-
trary element o f
Given two cardinals a and b , the ordering of less than or equal to, or greater
than or equal to, means that every s e t of cardinal a is subpotent w i t h every
s e t of cardinal b . Obvious definition of s t r i c t ordering; notations 6 , < .
Card a . T h i s i s only a minor inconvenience i n the definition.
5.5. CARDINAL SUM, CARDINAL PRODUCT AND EXPONENTIATION
Let a and b be cardinals; the cardinal sum a + b i s defined as the cardinal
of the union of two disjoint sets of cardinal a , respectively b . We denote
the cardinal sum by + (boldface) t o avoid confusion w i t h the ordinal sum +
in 3.1. Thus we can identify, in 5 6 below, Card w w i t h W i t s e l f , and write
a +1 = W and y e t W + ~ > L S .To be rigorous, we should also distinguish bet-
ween the ordering relation for cardinals and for ordinals. In practice the context
will always permit the distinction. Since cardinal multiplication and exponentia-
tion are denoted by a x b and ab (notations from 1.2 and 1.6), there will be
no confusion w i t h the operations of ordinal multiplication and exponentiation
a.b and ba . In particular the cardinal notation “ i s not necessary: ~3~
will be sufficient.

The sum a + b does not depend upon the choice o f d i s j o i n t sets o f cardinal
and cardinal b . Cardinal addition i s commutative and associative. We have
a
a + O = a . Finally a b a ' and b & b ' imply a+b,( a ' + b ' .
The cardinal product a x b i s defined as the cardinal o f the Cartesian product
of a set o f cardinal a with a set o f cardinal b . There w i l l be no inconvenience
i n using the same symbol f o r cardinal m u l t i p l i c a t i o n and f o r the Cartesian product
of two sets (see 1.2). The cardinal product does not depend upon the choice o f the
sets o f cardinal a , resp. b . Cardinal m u l t i p l i c a t i o n i s commutative, associa-
tive, and distributive over cardinal addition:
We have a % 0 = 0 and a x 1 = a . F i n a l l y a G a ' and b,<b' imply a x b s a ' x b '
The cardinal power ab i s defined as the cardinal o f the power between sets o f
cardinal a , resp. b (notation from 1.6). Cardinal exponentiation does not de-
pend upon the choice o f the sets o f cardinal a , resp. b .
We have 'a = 1 , 'a = a , aO = 0 f o r a # 0 , and al= 1 . Moreover f o r
b # 0 , conditions a,< a' and b,< b' imply ab C a'b' .
Finally the equipotences indicated i n 1.6 become cardinal equalities:
(b+c)a = (ba)$(Ca) ; then '(ash) = ('a)x('b) ; and C(ba) = ( b * c ) a .
(a+b)%c = ( a x c ) + ( b x c ) .
§ 6 - ALEPH, HARTOGS, ALEPH RANK
6.1. ALEPH
I n the case o f a set a which has an ordinal equipotent t o it, we take as the
definition o f the cardinal o f a , denoted s t i l l by Card a , the smallest ordinal
equipotent t o a . In s p i t e o f the very d i f f e r e n t notion o f cardinal as defined
i n 5.4, t h i s new Card a b i j e c t i v e l y corresponds t o the o l d notion, a t l e a s t f o r
sets a which are equipotent t o an ordinal. Such sets are called well-orderable
i n ch.2 5 2.5 below. The cardinal o f such a set i s called an aleph. Definitively,
we have the following a r t i f i c i a l but general and rigorous definition: i f a i s
equipotent t o an ordinal, then the smallest such i s Card a ; otherwise Card a
i s the set o f a l l sets o f minimum fundamental rank which are equipotent with a .
Another d e f i n i t i o n o f aleph, which i s equivalent t o the preceding one: an aleph i s
an ordinal o( which i s equipotent t o no ordinal < o( (less than with respect t o
the ordering o f the ordinals). I n particular, the f i n i t e alephs are the integers,
the f i r s t i n f i n i t e aleph i s 0 .
We w i l l see i n ch.2 5 2.5 that, with the axiom o f choice, every cardinal i s an
aleph (equivalently every set i s well-orderable).
Notice that i f o( and are two equipotent ordinals, then every intermediate
ordinal i s equipotent t o them. Moreover, f o r every i n f i n i t e ordinal a( , the
-

Chapter 1 21
successor o( +1 i s equipotent w i t h iA . It follows t h a t every i n f i n i t e aleph i s
a l i m i t o r d i n a l .
6.2. HARTOGS SET, OR HARTOGS ALEPH
Let a be an i n f i n i t e set. We say t h a t an o r d i n a l u i s i n j e c t a b l e i n a , i f f
there e x i s t s an i n j e c t i o n o f u i n t o a , o r equivalently i f a subset o f a i s
equipotent w i t h u . I f u i s i n j e c t a b l e i n a , then every ordinal s u and
every o r d i n a l equipotent w i t h u i s i n j e c t a b l e i n a . Since a i s i n f i n i t e ,
every i n t e g e r i s i n j e c t a b l e i n a . However i n order t h a t c3 be i n j e c t a b l e i n a ,
i t i s necessary t h a t a be Dedekind-infirite: see 2.6.
Given a s e t a , the o r d i n a l s i n j e c t a b l e i n a form a set.
0 For each o r d i n a l u and each i n j e c t i o n f o f u i n t o a , t h i s f , and conse-
quently u , i s defined by t h e s e t o f ordered p a i r s ( f x , f y ) f o r which x < y < u .
Such a s e t i s a r e l a t i o n , i n t h e sense o f ch.2 5 1 below; and a l l these r e l a t i o n s
form a s e t by the axioms o f ZF. 0
Hence, the s e t o f a l l o r d i n a l s i n j e c t a b l e i n a given s e t i s an aleph, which we
shall c a l l the Hartogs set, o r t h e Hartogs aleph o f a . This i s also t h e smallest
ordinal which i s n o t i n j e c t a b l e i n a (HARTOGS 1915).
a
6.3. SUCCESSOR ALEPH, LIMIT ALEPH
I f a i s i t s e l f an aleph, then t h e Hartogs aleph o f a i s t h e unique aleph imme-
d i a t e l y greater than a , i n t h e sense t h a t there i s no s t r i c t l y intermediate o r d i -
nal ( w i t h respect t o subpotence) between a and i t s Hartogs aleph. We shall denote
the l a t t e r by a+ and c a l l i t the successor aleph o f a .
For example
countable ordinals; o r again t h e l e a s t uncountable o r d i n a l . L e t t i n g ,
f o r each i n t e g e r i we l e t i+l= ( (3 i)+ .
The union o r supremum o r d i n a l o f an a r b i t r a r y s e t o f alephs i s again an aleph. For
example, from the preceding W i ( i integer), we l e t W U = Sup( w i ) , which i s
an aleph. We c a l l a non-successor aleph, such as &J o r Urn , a l i m i t aleph.
w 1 = LJ+ denotes the successor aleph o f w , and i s the s e t o f a l l
W o =
6.4. ALEPH RANK
We generalize the preceding notation. Given an a r b i t r a r y ordinal u , f o r an i n f i -
n i t e aleph a we w r i t e a = W u i f there e x i s t s a f u n c t i o n f w i t h domain u+l
(the successor o r d i n a l o f u ), such t h a t t h e i n i t i a l ordered p a i r (0, W ) and
the f i n a l ordered p a i r (u,a) belong t o f ; and such that, i f the ordered p a i r
(x,y) belongs t o f w i t h x an o r d i n a l < u and y an aleph, then (x+l,y+)
belongs t o
form (x,y) with Sup x ,C u then the ordered p a i r (Sup x, Sup y) belongs
t o f .
f ; and f i n a l l y such t h a t i f f contains a s e t o f ordered p a i r s o f the

Thus f o r each o r d i n a l
every i n f i n i t e aleph a there e x i s t s a unique o r d i n a l u such t h a t a = O u .
We c a l l u the aleph rank o f a . We have u ,< W ; e q u a l i t y i s possible: see
f o r instance ch.2 5 6.6.
For every o r d i n a l u we have Wu+l = ( G )u)+ , the successor aleph o f (3 .
Moreover, f o r every s e t o f o r d i n a l s x , we have t h e supremum e q u a l i t y :
u , there e x i s t s a unique aleph G,, . Conversely f o r
(SUP x) .sup wx = w
Hence an i n f i n i t e aleph i s a successor o r a l i m i t aleph, according t o whether i t s
aleph rank i s a successor ordinal o r l i m i t o r d i n a l ( i n c l u d i n g 0, since a,,= id).
It follows from t h e correspondence between alephs and aleph ranks t h a t , given a
s e t o f alephs ai and an aleph b , i f a i 4 b f o r a l l ai then Sup ai 6 b .
6.5. I n the presence o f t h e axiom o f choice, t h e continuum hypothesis i s equiva-
l e n t t o t h e e q u a l i t y &2 = O1 . However, w i t h the axioms o f ZF alone i n the
absence o f the axiom o f choice, the e q u a l i t y L32 = c d 1 i s , a p r i o r i , a stronger
assertion than the continuum hypothesis. Indeed, there may e x i s t a model o f ZF
without choice, where there i s no s t r i c t l y intermediate s e t ( w i t h respect t o sub-
potence) between G) and "2 , y e t where W 1
The s i t u a t i o n i s d i f f e r e n t w i t h the generalized continuum hypothesis, which implies
the axiom of choice (see ch.2 exerc.1). Thus the generalized continuum hypothesis
implies t h e e q u a l i t y a2 = a+ f o r each i n f i n i t e aleph a .
However, i t seems possible t o construct a model o f ZF s a t i s f y i n g
every i n f i n i t e aleph
thus negating the axiom o f choice.
Elementary properties concerning addition, m u l t i p l i c a t i o n and exponentiation of
i n f i n i t e alephs
ch.2 5 3.8 t o 3.10.
i s incomparable w i t h & 2 .
a2 = a+ f o r
a , and y e t having non-aleph cardinals which are incomparable,
w i l l be obtained with the help o f r e l a t i o n s and isomorphisms, i n
§ 7 - FILTER, ULTRAFILTER AXIOM
7.1. FILTER
,--
Given a s e t a , r e c a l l t h a t a f i l t e r on a i s a s e t f o f non-empty subsets o f
a , such t h a t
( 1 ) i f ~ € 5and x s y s a , t h e n Y E T ;
(2) i f x,y E , then the i n t e r s e c t i o n x n y E F;thus every f i n i t e i n t e r -
section o f elements o f F i s an element o f F .
Forhxample, the s e t o f complements o f f i n i t e subsets o f k) i s a f i l t e r on c.3
Every i n t e r s e c t i o n o f f i l t e r s on a i s a f i l t e r on a .
L e t be a f i l t e r on a , and b 5 a such t h a t b n x i s non-empty f o r every

Chapter 1 23
element x of LT. Then the set of intersections b n x constitutes a f i l t e r on
b , cal'led the f i l t e r induced by on b .
Let & be a set of subsets of a for which every finite intersection of i t s ele.
ments i s non-empty. Then the set of supersets of these intersections constitutes
a f i l t e r on a , called the f i l t e r generated by h! .
Let 3 , 3 be two f i l t e r s on the same set. 9
t o extend 3 , i f i t includes ; strictly finer i f i t strictly includes F .
Given a f i l t e r on a and b C_ a , either b E or a-b E , or every
intersection of any x 6 with b and with a-b i s non-empty. Hence there
exists a f i l t e r finer than 7 which contains
which contains a-b .
Let a be a set. If a set of f i l t e r s on a i s totally ordered by the comparison
relation "finer than", or more generally i f this comparison relation i s a direc-
ted partial ordering (i.e. given two f i l t e r s , there i s a third f i l t e r which i s
finer than b o t h ) , then the union of the f i l t e r s i s a f i l t e r on a .
i s said t o be finer than 3 , or
b , or a f i l t e r finer than 3
7.2. ULTRAFILTER, ULTRAFILTER AXIOM
Given a set a , an ultrafilter on a i s a f i l t e r for which there i s no strictly
finer f i l t e r on a . For example i f u E a , then the set of subsets of a con-
taining u i s an ultrafilter, said t o be trivial.
Already for the set w , the axioms of ZF alone are not sufficient t o prove the
existence of a non-trivial ultrafilter. I t i s necessary t o add, for instance, the
ultrafilter axiom (also called boolean prime ideal axiom), which asserts that
for every set a and every f i l t e r on a , there exists an ultrafilter on a
which i s finer t h a n 9 . For example this implies the existence of an ultrafilter
on 0 which contains as elements all complements of finite subsets of w .
We will see in ch.2 5 2.8 that the axiom of choice implies the ultrafilter axiom.
For a model of ZF having no ultrafilters other than the trivial ultrafilters,
see BLASS 1977.
A necessary and sufficient condition that a f i l t e r 7 on a be an ultrafilter,
i s that for every subset x o f a , either x E 7 or a-x e 3 .
Let be an ultrafilter and-x E F .Then for every partition of x 1%
finite number o f disjoint subsets, one and only one of these subsets belongs-
Every f i l t e r 7 on a i s the intersection of all ultrafilters on a which are
finer than 7 (uses the ultrafilter axiom).
To calculate the number of f i l t e r s on a set, see ch.2 exerc. 2.
To see the impossibility of countably generating an ultrafilter on CS, see
ch.2 5 8.1.
-
-
t.3.

§ 8 - TOPOLOGYON SETS OF INTEGERS
S t a r t w i t h t h e s e t N o f the natural integers. For each ordered p a i r o f f i n i t e
sets F, G o f N , l e t UF denote the s e t o f those subsets o f N which include
F and are d i s j o i n t from G . For F and G empty, we obtain t h e e n t i r e s e t Y ( N )
Note t h a t UF i s non-empty i f f F and G are d i s j o i n t .
The i n t e r s e c t i o n UF n U:: i s U~~~~
any union o f preceding U sets, then the i n t e r s e c t i o n o f any two open sets i s
s t i l l an open set; so t h a t we obtain a topology on T (N) .
The complement o f a U s e t i s a union o f U sets, thus an open set; so t h a t each
U s e t i s both open and closed, i.e. the complement o f an open set; more b r i e f l y
each U i s a clopen set.
This topology i s Hausdorff: given two subsets A and B o f N , supposed t o be
d i s t i n c t , take a f i n i t e s e t F which i s included i n A y e t not i n B , and a
f i n i t e s e t G included i n B y e t n o t i n A , w i t h F and G n o t both empty:
then UF and UG separate A from B .
G
G
. Consequently, d e f i n i n g an o p e n s e t t o be
G F
8.1. CONVERGENT SEQUENCE, CLOSURE
Consider an a-sequence o f subsets Hi o f N ( i natural integer). We say t h a t
t h i s sequence converges, and t h a t a subset H o f N i s the limit o f t h e Hi , i f
f o r every i n t e g e r x , e i t h e r x belongs t o H and then x belongs t o Hi from
some index on ; o r x belongs t o the complement o f H and then x belongs t o the
complement o f Hi from some index on.
The closure o f a s e t
l i m i t elements f o r a l l convergent ra-sequences o f elements o f S . This closure
i s also the smallest closed superset o f S , w i t h respect t o inclusion.
S o f sets o f integers, i s defined as being t h e s e t of a l l
8.2. THE TOPOLOGY I S COMPACT
I n other words, if T ( N )
by a f i n i t e number o f these sets.
0,Suppose we have an a-sequence o f ordered p a i r s (F(i),G(i)) (iinteger) of
f i n i t e subsets o f N , and l e t Ui = UG(!) . Assume t h a t f o r each i , the union
Uo u U1 u ... u U. i s s t r i c t l y included i n Q(N) . We w i l l show t h a t t h e union
o f a l l these Ui i s d i s t i n c t from T(N) .
We see t h a t there e x i s t s a f u n c t i o n h which, t o each i , associates an element
hi o f F ( i ) u G ( i ) , which can be assumed t o be non-empty. Giving hi t h e s i g n
(+) o r (-) according t o whether i t belongs t o F ( i ) o r G ( i ) , we can choose h
t o v e r i f y the f o l l o w i n g condition. For each i , there e x i s t s an element i n the
difference T ( N ) - (Uo u ... u Ui) , which contains as an element a l l those
hl,...,hi
i s covered by a union o f open sets, then i t i s covered
F(1)
1
ho,
hj
o f s i g n (-), and none o f sign (+). It follows t h a t the i d e n t i t y
hi =

Chapter 1 25
( w i t h i # j) implies t h a t hi and h . have the same sign. F i n a l l y the s e t o f a l l
hi o f s i g n (-), does n o t belong t o the union o f the Ui f o r a l l integers i . 0
It follows immediately from t h i s compactness, t h a t t h e clopen sets are e x a c t l y the
f i n i t e unions o f the preceding U sets.
J
8.3. Consider an a-sequence o f ordered p a i r s (F(i),G(i)) o f f i n i t e subsets o f N
( i integer), and assume t h a t F ( i ) and G ( i ) are both non-empty and d i s j o i n t , f o r
each i . L e t h be a f u n c t i o n which, t o each i , associates an element hi i n
F ( i ) u G ( i ) ; w i t h the c o n d i t i o n t h a t i f hi = h . (i# j) , then e i t h e r hi
both t o F ( i ) and F ( j ) , o r both t o G ( i ) and G ( j ) . Then we associate the sign
(t) o r (-) t o hi , according t o whether i t belongs t o F ( i ) o r t o G ( i ) .
Let Vh be the open s e t formed by t h e subsets X o f N f o r which there e x i s t s an
i such t h a t hi has s i g n (+) and hi E X , o r hi has sign (-) and hi X .
Then the open set, union o f the UG(i) , i s t h e i n t e r s e c t i o n o f the Vh f o r a l l t h e
functions h previously defined.
belongs
J
F ( i )
8.4. DENSE SET, BAIRE'S CONDITION
A set i s c a l l e d dense,i f i t s i n t e r s e c t i o n w i t h every non-empty open s e t i s non-
empty.
For every function h associating t o each i n t e g e r i an element hi o f N ,
taking i n f i n i t e l y many values hi w i t h any sign, the open s e t Vh previously
defined i s dense. Thus there e x i s t i n f i n i t e l y many dense open sets, although t h e
only dense closed s e t i s 9 (N) .
Notice t h a t the i n t e r s e c t i o n o f two dense open sets i s a dense open set.
Every compact topology s a t i s f i e s BAIRE's condition: every countable i n t e r s e c t i o n
o f dense open sets i s non-empty, and even dense. We l e t the proof t o the reader,
f o r instance by t a k i n g a f i n i t e s e t o f dense ooen sets; and then i n f i n i t e l y many.
0 Let us sketch a d i r e c t proof f o r the present topology. Every open s e t i s a union
o f sets UF , thus by 8.3, a countable i n t e r s e c t i o n o f open sets
Let us prove BAIRE's c o n d i t i o n f o r the dense Vh , hence those corresponding t o
the h t a k i n g i n f i n i t e l y many values. To do t h i s , replace each Vh , by reducing
h t o only those values w i t h s i g n (t), o r only those o f sign (-), assumed t o be
i n f i n i t e i n number. Denote by W the i n f i n i t e s e t o f these values. The open s e t Vh
i s reduced t o the s e t o f subsets X o f N such t h a t W n X i s non-empty, o r t o
the s e t o f subsets X such t h a t W n (N-X) i s non-empty. We s h a l l see i n ch.2
5 8.1 (countable case), t h a t there e x i s t s a s u i t a b l e s e t X f o r a l l the W , w i t h
W n X and W n (N-X) i n f i n i t e . Thus there e x i s t s a s e t X i n c l u d i n g any given
f i n i t e s e t and excluding any given f i n i t e s e t ( d i s j o i n t each from the other);
hence there e x i s t s an
G
Vh .
X belonging t o any given open set. 0

I n conclusion, an open set defined by an a-sequence (F(i),G(
f o r every ordered p a i r o f f i n i t e d i s j o i n t subsets F, G o f N
i with F ( i ) d i s j o i n t from F and G(i) d i s j o i n t from G .
) ) i s dense i f f ,
there exists an
§ 9 - NATURAL SUM AND PRODUCT FOR ORDINALS
From the unique decomposition o f an ordinal i n t o a sum o f decreasing powers o f 0 ,
one defines the commutative operations o f natural sum and natural product f o r
ordinals: t h i s goes back t o HESSENBERG 1906 ; see also BACHMANN 1967 p. 107.
For the sum, we begin with O( = w0( (l).ml + ... t W w(h).mh ,where the
coefficients ml, ...,mh are integers, and where cx (1) 7 ... 7 o( (h) are
ordinal exponents, and p = Wo((l).nl + ... + G a ( h ) . n h . We can always assume
t h a t the ordinal exponents are the same f o r both decompositions, i f necessary by
inserting terms with c o e f f i c i e n t zero.
Then the natural sum o( @ fs i s defined as:
c3 M(l).(ml+nl) + ... + ~ c ) ~ ( ~ ) . ( m ~ + n ~ ).
For the natural product, one f i r s t defines the product o f u' and U'as
being *@ . Then f o r two a r b i t r a r y ordinals, each written i n the form o f
a sum o f powers o f G) , one multiplies them as with polynomials.
With respect t o these notions, one defines the "relative" ( i . e . positive o r nega-
t i v e ) ordinals, by substituting i n the coefficients
integers. Then the rational ordinals by taking the quotient f i e l d : see SIKORSKI
1948, and also BACHMANN 1967. More d i f f i c u l t and less "natural", one can define
t r a n s f i n i t e reals; see f o r instance KLAUA 1959 and 1960.
m , positive and negative
EXERCISE 1 - CONSISTENCY OF THE AXIOM OF FOUNDATION
To see t h a t the addition o f the axiom o f foundation does not imply a contradiction,
i f the other axioms o f ZF are consistent, we define an ordinal t o be a set not
only t r a n s i t i v e and t o t a l l y ordered by the membership relation, but also satis-
fying the following foundation condition. For every non-empty subset
ordinal, there exists an element y o f , x which i s d i s j o i n t from x : see
BERNAYS 1968 p. 80.
Next, one must redefine an integer as a f i n i t e ordinal. Then say t h a t a set a i s
well-founded i f every sequence xi indexed by integers, with xo = a and xi+l
belonging t o xi f o r each i , i s f i n i t e . Finally, one v e r i f i e s t h a t the well-
founded sets with the membership relation, s a t i s f i e s a l l the axioms o f ZF, inclu-
ding the axiom o f foundation.
N.B. It i s not s u f f i c i e n t t o define a well-founded set (other than an ordinal) by
x o f an

Chapter 1 21
the f i r s t foundation c o n d i t i o n
EXERCISE 2 - ALEXANDROFF-FODOR THEOREM
A subset o f
ifi t i s equipotent w i t h
element i n the subset.
We say t h a t a function f w i t h domain
every x i n Dom f .
1 - I f f i s regressive, then there e x i s t s a countable ordinal o( f o r which the
set o f x such t h a t f x = d i s c o f i n a l (ALEXANDROFF 1935, countable axiom o f
choice used). S t a r t i n g w i t h a countable ordinal o( (0) , i f f o r every /s < o< (0)
there are countably many x where f x = fi , then there e x i s t s an m ( l ) 7 a( (0)
such t h a t from t h a t p o i n t on,
sequence O( ( i ) ( i integer), and so 8 = Sup O( ( i ) i s countable by 2.5 (tounta-
ble axiom o f choice), and s a t i s f i e s
veness o f f .
2 - We say t h a t a s e t o f ordinals i s closed i f , f o r every s t r i c t l y increasing
w -sequence o f elements, the supremum ordinal belongs t o the set. Clearly, the
intersection o f two closed c o f i n a l sets i s closed c o f i n a l . Also the i n t e r s e c t i o n
o f a countable s e t o f closed c o f i n a l sets i s closed c o f i n a l : t h i s reduces t o an
0 -sequence which i s decreasing w i t h respect t o inclusion.
We say t h a t a s e t i s stationary, i f i t i n t e r s e c t s every closed c o f i n a l set. Note
that every stationary s e t i s c o f i n a l .
If f i s regressive, then there e x i s t s a countable ordinal o( f o r which the set
o f x such t h a t f x = o( , i s s t a t i o n a r y (FODOR 1966, uses axiom o f choice).
Suppose the contrary. Since f o r every countable ordinal /?J , t h e s e t o f x
(fx = /?I ) i s n o t stationary, there e x i s t s a closed c o f i n a l s e t C ,-,(axiom
of choice) on which f x # /5. S t a r t i n g w i t h an a r b i t r a r y countable ordinal w.(0),
take d(1)> 4 ( 0 ) where 4 (1) belongs t o n C ,-,( /5 < o((0)) . I t e r a -
t i n g t h i s , we obtain an w-sequence o( (i)(iinteger) , and thus 21 = Sup o( ( i ) .
Hence belongs t o A C ( /?J < 6 ) , and so f 8 i s d i s t i n c t from every
/s < ‘E( , and so f y >, , which contradicts the regressiveness o f f .
m 1, the s e t o f countable ordinals, i s s a i d t o be c o f i n a l i n W ,
w1 , so t h a t every element o f U 1 admits a greater
W 1 - { O } i s regressive, i f f x < x f o r
f x >/ O( (0) . I t e r a t i n g t h i s , we obtain an GO-
f 8 >/ , thus c o n t r a d i c t i n g the regressi-
3 - Extend t h i s proposition t o t h e case where Dom f i s stationary.
EXERCISE 3 - A CLASSICAL INTERPRETATION OF THE ORDINAL EXPONENTIATION
Let o( , fi be two ordinals. Consider functions f with domain /3 , taking
values i n o( . We say t h a t such an f i s almost zero i f f ( i ) # 0 f o r a t most
f i n i t e l y many elements i o f /3 . Given two almost zero functions f and g ,
l e t f < g i f there e x i s t s an i i n (3 w i t h f ( i )< g ( i ) i n the usual ordering
f o r ordinals, and f ( j )= g ( j ) for a l l j such t h a t i< j < b . L e t f< g if

f < g o r f = g . I n other words, the s e t o f such almost zero functions
by l a s t difference.
1 - Show t h a t 6 i s a well-ordering on the s e t o f almost zero functions
reader i s assumed t o know t h i s notion; otherwise see ch.2 5 2.4).
s ordered
the
2 - Show t h a t t h i s well-ordering i s isomorphic w i t h the exponential o(" (induc-
t i o n on (3 ).
EXERCISE 4 - A FAMILY OF SUBSETS I N A DENUMERABLE SET, OR I N THE CONTINUUM
1 - Given a denumerable s e t E , show t h a t there e x i s t continuum many denumerable
subsets A o f E , having pairwise a f i n i t e intersection. Indeed take f o r E the
s e t o f a l l f i n i t e sequences w i t h possible values 0 and 1 . Then define each A
by an w-sequence o f 0 and 1 ; say t h a t a given f i n i t e sequence belongs t o A
i f f i t i s an i n i t i a l i n t e r v a l o f t h e W-sequence associated w i t h A ( t h e n o t i o n
o f i n i t i a l i n t e r v a l i s obvious; i f necessary see ch.4 § 2.1).
2 - Modulo the axiom o f choice, complete t h e given family o f sets
t o s a t i s f y the f o l l o w i n g condition: f o r every denumerable subset X o f E ,
there e x i s t s an A such t h a t t h e i n t e r s e c t i o n A n X be i n f i n i t e .
3 - Now consider a s e t E w i t h continuum c a r d i n a l i t y . Show t h a t , modulo the
axiom o f choice, there e x i s t (2 t o the power al) many subsets A o f E , each
having c a r d i n a l i t y c.dl , and having pairwise a countable i n t e r s e c t i o n . Indeed
take f o r E the s e t o f a l l countable ordinal-indexed sequences o f 0 and 1 .
Then define each A by an W1-sequence o f 0 and 1 ; say t h a t a given countable
sequence belongs t o A i f f i t i s an i n i t i a l i n t e r v a l o f the ul-sequence asso-
c i a t e d w i t h A (note t h a t t h e axiom o f choice i s needed t o see t h a t the s e t E
o f a l l countable sequences i s equipotent w i t h the continuum).
4 - Complete the given f a m i l y o f sets A i n order t h a t , f o r every subset X o f
E having c a r d i n a l i t y O1 , there e x i s t s an A such t h a t t h e i n t e r s e c t i o n
A X have c a r d i n a l i t y cS1 .
A i n order

29
CHAPTER 2
RELATION, PARTIAL ORDER1NG, CHAIN, ISOMORPHISp1, COFIWALITY
§ 1- RELATION, MULTIRELATION, RESTRICTION, EXTENSION, COHERENCE
LEMMA, AXIOM OF DEPENDENT CHOICE
Let E be a s e t and n an integer. In ch.1 5 2.3 we defined the notion o f
n-tuple w i t h values i n E . We s e t aside two elements c a l l e d values, which are
denoted + and - ( f o r instance, these can be defined by 0 and 1). An n-ary
relation. w i t h e E , o r based on E , i s a f u n c t i o n R which associates the
value R(xl ,...,xn) = + o r - t o each n-tuple x1 ,...,xn i n E ( f o r conve-
nience, we o f t e n denote t h e n-tuple by i t s indices 1 t o n instead o f from 0
t o n-1 ) . The s e t E , the base o f R , w i l l be denoted I R I . The i n t e g e r n
w i l l be c a l l e d the * o f R . For n = 1,2,3, we w i l l say a unary, binary,
ternary r e l a t i o n .
For n = 0 , we adopt the convention t h a t there e x i s t two 0 - 9 r e l a t i o n s based
on E , which we denote by (E,t) and (E,-) : the 0-ary r e l a t i o n s w i t h value t
and value - .
We adopt the convention t h a t , f o r each p o s i t i v e n , there e x i s t s a unique n-ary
r e l a t i o n w i t h empty base. However, there e x i s t two 0-ary r e l a t i o n s w i t h empty
base: (O,+) and (0,-) . These conventions agree w i t h the c a l c u l a t i o n o f the
n
number o f n-tuples w i t h values taken from a base o f finit'e cardinal p ; i . e . p .
The number o f n-ary r e l a t i o n s based on p elements i s "2 t o the power pn 'I. Here
ordinal exponentiation coincides w i t h cardinal exponentiation np , f o r n and p
f i n i t e .
Examples o f r e l a t i o n s . The usual ordering o f the integers i s the r e l a t i o n R
which s a t i s f i e s R(x1,x2) = + when x1 x2 and - when x1 > x2 . A group i s
a ternary r e l a t i o n t a k i n g t h e value t when x1.x2 = x3 and t h e value - when
x1.x2 # x3 , where . i s the composition law o f t h e group. Instead o f x1,x2,x3
we s h a l l o f t e n use x,y,z .
A m u l t i r e l a t i o n w i t h base E i s a f i n i t e sequence R o f r e l a t i o n s R1, ...,Rh
(h integer), each w i t h base i s c a l l e d a component
o f the m u l t i r e l a t i o n R . We c a l l t h e arity o f R t h e sequence (nl, ...,nh) o f
a r i t i e s o f the components R1, ...,Rh . We say then t h a t t h e m u l t i r e l a t i o n R i s
(nl, ...,n h ) - x . The l e n g t h h o f the sequence o f indices can be zero: i n t h i s
case, the m u l t i r e l a t i o n i s reduced t o i t s base E . Instead o f the notation R1,
R2,R3 , o f t e n we s h a l l use R,S,T . I n t h e case where h = 2 , we w i l l say
-
E . Each Ri (i= 1, ...,h)

a birelation; f o r h = 3 a t r i r e l a t i o n , etc. Finally, the base o f a m u l t i r e l a t i o n
R shall be denoted I R I .
Example. An ordered group i s a (3,2)-ary b i r e l a t i o n which i s formed o f the ternary
group r e l a t i o n and the binary ordering relation.
A r e l a t i o n o r m u l t i r e l a t i o n w i l l be called f i n i t e , i n f i n i t e , countable
o r continuum-equipotent, according t o whether i t s base i s f i n i t e ,
countable, denumerable or continuum-equipotent. The cardinal o f the mu
R i s the cardinal o f i t s base J R I .
denumera-
i n f i n i t e ,
t ire1a tion
Given two multirelations R, S with common base E , we c a l l the concatenation
o f R and S , denoted (R,S) , the sequence o f components o f R followed by
the components o f S , i n which case f o r the l a t t e r the indices are increased by
the number of terms i n R .
1.1. n-ARY RESTRICTION, n-ARY EXTENSION
Let R be an n-ary r e l a t i o n with base E , and l e t F be a subset o f E . We
c a l l the n - a 3 r e s t r i c t i o n o f R t o F , denoted by R/F , the n-ary r e l a t i o n
taking the same value f o r each n-tuple with values i n F . The notion o f r e s t r i c -
t i o n o f a function i n ch.1 5 1.3, i s more general than t h a t o f n-ary r e s t r i c t i o n :
the former would allow one t o r e s t r i c t R t o an a r b i t r a r y subset o f the set 'E
o f n-tuples with values i n E , and not necessarily t o a subset o f the form 'F
with F S E . However i n practice, the context w i l l make the meaning o f the ad-
j e c t i v e "n-ary" obvious: we t a c i t l y assume t h i s .
For the a r i t y 0, the r e s t r i c t i o n t o F o f the 0-ary r e l a t i o n (E,+) w i l l be
(F,+) ; s i m i l a r l y with - ; t h i s remains v a l i d f o r empty F .
Given a r e l a t i o n R w i t h base E and a superset E+ o f E ,we c a l l an 9-
-sion o f R t o E+ any r e l a t i o n with base E+ whose r e s t r i c t i o n t o E i s R .
Let R, R' be two n-ary relations w i t h common base E . If f o r every subset
X 3 E with cardinal 6 n , we have R/X = R ' / X ,then R = R' .
Given a m u l t i r e l a t i o n R = (R1,-..,Rh) with base E and a subset F o f E ,
we define the r e s t r i c t i o n o f R t o F , denoted by R/F , t o be the multirela-
t i o n (R1/F, ...,Rh/F) . Given a m u l t i r e l a t i o n R with base E and a superset
E+ o f E ,we c a l l an extension o f R t o Ef any m u l t i r e l a t i o n with base E+
whose r e s t r i c t i o n t o E i s R . Equivalently, any sequence (R;,.. .,Rh) where
each Ri
Let R, R' be two multirelations o f common a r i t y (nl, ...,nh) and with common
base E . I f f o r each subset X o f E with cardinal 6 Max(nl, ...,nh) ,
have R/X = R'/X , then R = R' .
+
+ i s an extension o f Ri t o E+ ( i = 1, ...,h) .
-
- -

[Slfm 118] theory of relations roland fraisse (nh 1986)(t)

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