COMBINATORIAL
SET THEORY:
PARTITION RELATIONS
FOR CARDINALS
PAUL ERDOS,
ANDRAS HAJNAL,
ATTILA MATE,
RICHARD RADO
1984
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM 0 NEW YORK 0 OXFORD
0AKADCMIAI KIAD6, BUDAPEST
1984
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Main entry under title:
Combinatonal set theory.
(Studies in logic and the foundations of mathematics; v. 106)
Bibliography: p.
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PREFACE
Ramsey’s classical theorem in its simplest form, published in 1930,says that if
we put the edges of an infinite complete graph into two classes, then there will be
an infinite complete subgraph all edges of which belong to the s v e class. The
partition calculus developed as a collection of generalizations of ‘this theorem.
The first important generalization was the ErdBs-Dushnik-Miller theorem
which says that for an arbitrary infinite cardinal K , if we put the edges of a
complete graph of cardinality K into two classes then either the first class
contains a complete graph of cardinality K or the second one contains an infinite
complete graph. An earlier result of Sierpinski says that in case K = 2’0 we cannot
expect that either of the classes contains an uncountable comp1e;e graph. The
first work of a major scope, which sets out to give a ‘calculus’ of partitions as its
aim, was the paper “A partition calculus in set theory” written by ErdBs and
Rado in 1956. In 1965, ErdGs, Hajnal, and Rado gave an almost complete
discussion of the ordinary partition relation for cardinals under the assumption
of the generalized continuum hypothesis. At that time there were hardly any
general results for ordinals, though there were some results of Specker, and the
paper of ErdBs and Rado quoted above also contains some results for them. The
situation has now changed considerably. The advent of Cohen’s forcing method,
and later Jensen’s theory of the constructible universe gave new spurs to the
development of the partition calculus. The main contributors in the next ten
years were J. E. Baumgartner, C. C. Chang, F. Galvin, J. Larson, R. A. Laver,
E. C. Milner, K. Prikry, and S. Shelah, to mention but a few. Independence
results are beyond the scope of this book, though it will occasionally be useful to
quote some of them in order to put theorems of set theory into their real
perspective. An attempt t o give a survey that deals also with independence results
was made by ErdGs and Hajnal in their paper “Solved and unsolved problems in
set theory” which appeared in the Tarski symposium volume in 1974. The
progress here is, however, so rapid that this survey was obsolete to a certain
extent already when it appeared in print.
Our aim in writing this book is to present what we consider to be the most
important combinatorial ideas in the partition calculus, and we also want to give
a discussion of the ordinary partition relation for cardinals without the
6
PREFACE
assumption of the generalized continuum hypothesis; we tried to make this latter
as complete as possible. A separate section describes the main partition symbols
scattered in the literature. A chapter on the applications of the combinatorial
met hods in partition calculus includes a section on topology with Arhangel’skii’s
famous result that a first countable compact Hausdorff space has cardinality at
most the continuum, several sections on set mappings, and an account of recent
inequalities for cardinal powers that were obtained in the wake of Silver’s
breakthrough result saying that the continuum hypothesis cannot first fail at a
singular cardinal of uncountable cofinality. Large cardinals are discussed up to
measurability, in slightly more detail than would be necessary strictly from the
viewpoint of the partition calculus.
We assume some acquaintance with set theory on the part of the reader,
though we tried to keep this to a minimum by the inclusion of an introductory
chapter. The nature of the subject matter made it inevitable that we make some
demands on the reader in the way of mathematical maturity.
And we make another important assumption: the axiom ofchoice, that is the
axiomatic framework in this book is ZermebFraenkel set theory always with
the axiom of choice. There are interesting results in the partition calculus which
do not need the axiom of choice, but we have never made an attempt to avoid
using it. There are many interesting assertions that are consistent with set theory
without the axiom of choice but contradict this latter, and there are many
important theorems of set theory plus some interesting additional assumption,
e.g. the axiom of determinacy, that is known to contradict the axiom of choice.
We did not include any of these; unfortunate though this may be, we had to
compromise; we attempted to discuss infinity, but had to accomplish our task in
finite time.
CHAPTER I
INTRODUCTION
The notations generally used in this book are given in the first section, then
there follows a section listing the axioms of Zermelo-Fraenkel set theory. The
most important notions involving ordinals and cardinals are presented, and
finally such important tools in set theory as transfinite recursion, Mostowski’s
Collapsing Lemma, the Wellordering Theorem, Hausdorff s Maximal Chain
Theorem, Zorns’s Lemma, and Hausdorff s Cofinality Lemma are discussed.
1. NOTATION AND BASIC CONCEPTS
Although we shall rarely ever directly refer to any axiom of set theory below, a
natural, perhaps the most natural framework for the considerations in this book
is Zermel*Fraenkel set theory (denoted by Z F ) with the Axiom of Choice
(denoted by AC; the axiom system ZF AC is shortly written as ZFC). A list of
the axioms of ZFC is given in the next section. We shall make no effort to avoid
the use of AC even if this could be done, as it is beyond our aims to clarify the
exact role of this axiom in the topics discussed. We assume that the reader is
familiar with the basics of ZFC. Consjstency results are occasionally mentioned,
but their proofs are never given, so the reader need not be acquainted with the
Axiom of Constructibility (written as V = L ) and with forcing or the theory of
Boolean-valued models.
Following the convention introduced by J. von Neumann, we identified
ordinals with the set of their predecessors, and cardinals with their initial
ordinals. Ordinals are usually denoted by lower case Greek letters (excepting E, I,
o, u, cp, x, and $)with or without subscripts or superscripts. Among these, o has a
specific meaning (the least infinite ordinal), and the letters K, i,p , (T, and 5 always
denote cardinals, even if we d o not mention this explicitly. Cardinals and
ordinals may be denoted also by other letters if specifically mentioned. Integers,
which are defined as finite ordinals, are usually denoted by the lower case Roman
letters i, j, k, I, m, n, r, and s.
When writing logical formulas, we denote the logical connectives ‘not’, ‘and’,
‘or’, ‘implies’,‘if, ‘if and only if (this last one is occasionally abbreviated as ‘$.I”)
+
10
INTRODUCTION CH.
I
by 7, &, v , -, e,-in turn; the existential and universal quantifiers are written
as 3x (‘there is an x such that’) and Vx (‘for all x’), respectively. The language of
ZF (which is of course, the same as the language of ZFC), often called the
primitive language of ZF, contains, aside from these symbols, also the signs E
(membership or elementhood) and = (equality). Constants are not admitted in
the primitive language of ZF; variables (usually denoted below by x, y, z, u, . . .
with or without subscripts or superscripts) will occasionally be called set
variables, since in the intuitive interpretation they are meant to denote sets. As is
usual, we extend the primitive language of Z F by introducing certain constant
symbols and abbreviations commonly used in set theory. In this way, we shall use
the quantifier 3!x, meaning there is exactly one x, and the bounded (or restricted)
quantifiers Vx E y, 3x E y, and 3 !x E y. If cp(x) is a formula of ZF, then the classifier
{x: cp(x)) is intuitively interpreted as the class (or collection) of those sets x for
which cp(x) holds. The precise formal interpretation is a collection of rules about
how to eliminate a classifier from a formula whenever it occurs in it. According to
these rules, the formula y E {x: cp(x)) has to be replaced by the formula cp(y). If A
and B are two classes (or, rather, classifiers), then A = B is defined as Vx[x E A
o x E B], where x is a variable which does not occur (or;at least, is not free) in A
or B; this is an implicit reference to the Axiom of Extensionality, as generalized
for classes. The case when x is free in A or B (so that one must choose another
variable instead of x ) is called a clash (of variables). As any set x can be
considered the same as the class {y: y E x), this also explains the meanings of the
formulas x = A and A = x, where A is a class and x is a set. Finally, the formula
A E B, where A and B are classes, is interpreted as 3x [x = A & x E B ] , where x is a
variable not occurring in A or B. A class A is a real class if V x [ l x = A] (assume x
does not occur in A); in other words, A is a real class if and only if it is not (equal
to) a set. Note that if A is a real class then the formula A E B is false for any class B.
The classifier { x : x E y & cp(x)) always defines a set in virtue of the Axiom of
Replacement (see the next section); an often used shorter notation for this set is
{x E y: ~ ( x ) ) In
. Sections 1-4, classes will be denoted by upper case letters and
sets by lower case ones. From Section 5 on, classes will hardly ever be used, and
upper case letters will also denote sets unless the contrary is explicitly mentioned.
Further abbreviations added to the language of set theory are negations of
equality and membership ($1 and $, respectively; other binary predicates are also
often negated in this way, by crossing off), inclusion in the wider sense (E ; A S B
allows the equality of A and B),srrict (or proper) inclusion (c; equality is not
allowed), union of two classes (u),union of the elements of a class
dzference
of two classes (\), intersection of two classes (n),and intersection of the elements
of a class
We have n A = {x: Vy[y E A*x E y]), if clashes of variables are
avoided; hence, if A equals the empty set {x: x # x ) (denoted by the constant
(u),
(n).
NOTATION A N D BASIC CONCEPTS
I1
symbol 0) then ( ) A = {x: x = XI, this latter being a shorthand for the universal
class, or the class of all sets. x and y are called disjoint if x n y = 0. If x is a set,
then 9 ( x ) denotes the power set of x , i.e., .Y(x)= { y : y ~ x ) .
Singleton x is the one element set {xi, which can be defined as { y : y = x ' , . The
unorderedpair {xy:istheset (z:z=xvz=y).Incaseswhenwethinkit practical
we may define larger sets as well by enumerating its elements: {ab . . . c } or
{a, 6, . . ., c> denotes the set { x : x = a v x = b v . . . v x = c ) , where the dots . . .
are to be replaced by symbols according to a rule that should be straightforward (occasionally infinite sets are also written in this way; the last
classifier mentioned above is then, strictly speaking, meaningless, since it
contains a formula with infinitely many disjunctions, and there is no such
formula in ZF; but it will always be possible to replace it with a formula of ZF).
An ordered pair ( x y ) is defined as the set { ( x i ( x y ] ] ;a set z defined in some
other way from x and y so that one could, just by looking at z, tell what x and y
were would be equally good. A relation is a class of ordered pairs; for a relation R,
( x y ) E R is often written as x R y . If ( x y ) E R implies x , y E C for some class C,
then we say that R is a relation on the class C. The restriction R I A of a relation R
to a class A is defined as ,the relation { ( x y ) E R : x , y E A ) . (Given two classes A
and B, the symbol A IB will occasionally be used in other senses as well; it will
always denote a kind of restriction. This ambiguity of notation will always be
cleared up by the context.) An operation F is a relation such that, by looking at
the first element of a pair in F, one can always tell which the second element is, i.e.,
V x y z [ [ ( x y ) E F & (xz) E F ] - y = z ] (avoid clashes!). An operation F is one-toone or 1-1 if by looking at'the second element of a pair in F one can tell which the
firstelementis,i.e.,Vxyz[[(yx) E F &( z x ) E F l * y = z ] . IfFisanoperation that
is a set then F is called a function. For a class F, dom(F) denotes its domain,i.e.,
the class { y : 3 Z [ ( y Z ) E F ] } and ra(F) denotes its range, i.e., the class
{ y : 3Z[(Zy) E F ] } . These concepts have intuitive motivation only in case F is an
operation, but it will be usefulin formal arguments that we have defined them for
an arbitrary class F (cf. e.g. the proof of the Wellordering Theorem in Subsection
4.6 below).
IfF is an operation and x E dom (F),then the value of F at x , written as F(x), is
defined as the unique set y with ( x y ) E F. If x $ dom(F) then F ( x ) is defined as
the empty set Oin the strict formal sense; intuitively, however, it is better to think
that F(x) is not defined at all in this case. If F is an operation and X is a class then
the restriction of F to X is denoted by F P X , i.e., we put
F P X = { ( X ~ ) E FX: E X ) .
We shall also occasionally use Godel's notation
F"X
= ra(F P X ) (=
{ F ( x ) :x E Xndom(F))) .
12
INTRODUCTION CH.
1
Finally, if x and y are sets, then ,y (say: y pre x ) denotes the set of all functions
from x into y, i.e.,
x y = ( f : f i s a function, dom ( f ) = x , and ra ( j ’ ) z y i .
xy can be defined in exactly the same way in the case when y is a real class; we
shall, however, hardly ever be concerned with real classes; therefore here, and in
what follows, we shall be content with explaining the basic concepts for sets. The
~ f :x b y
notation f:x + y means that f is afunction from x into y, i.e., that , f “y.
means, on the other hand, that f is a function, x ~ d o m
( f ) ,and f ( x ) =y. Given
two functions f and g, their composition f o g is defined as the function h such
that h ( x ) = f ( g ( x ) )whenever x E dom (g)and g ( x ) E dom ( f ) ;in other words, we
Put
fog={(xy):
3zc(xz>Eg&(zy)Efl;.
A function is occasionally called an indexed set; in this case, its domain is called
the index set.
If F is an operation with domain D and F = { ( x z , ) : x E D),where z, is a set
somehow determined by x , then we may write F as ( z x : x E 0)(this latter is the
‘classifier’ for operations; note that its definition lacks mathematical precision,
and so it is only a visual device to inform the reader rather than a part of the
extended language of ZF).
A sequence is a function defined on an ordinal. A (usually finite) sequence
( x i : x,,=a, . . ., x k - = d ) may also be written as ( a , . . ., d ) (the commas may
be omitted). In case k = 2 the sequence ( a d ) of length two can be confused with
the ordered pair ( a d ) in this notation. In the strict formal sense, this confusion is
unpardonable, since these two are different sets. Intuitively, however, there is
usually no need to distinguish between them; in formal definitions, it is best to
think about (ad) as an ordered pair unless it is clear from the context that it
should be a sequence of length two.
If f is a function, then a choice function for f is a function g such that
dom (g)=dom (f)and g ( x ) E ~ ( x for
) every x E dom (1);
the Cartesian product
X f i s the set of all choice functions for f (in case f assumes the empty set 0 as a
value [at some element of its domain] then, obviously,
is also empty). The
Cartesian product X x Y of two classes X and Y is usually defined by
Xf
xx
Y={(uv):
U € X & U €
Y}.
Here ( u v ) is an ordered pair. Note that i f f is the function { ( O x ) , (ly)), where x
and y are arbitrary sets, then the Cartesian productxf is not the same as x x y,
though there is no harm in identifying the two in informal considerations. In
a similar way, the two (formally different) Cartesian products ( x x y ) x z and
THE AXIOMS OF ZERMELC+FRAENKEL SET THEORY
13
x x (y x z ) are also identified usually, and simply written as x x y x z. A matrix (or
double sequence) is a function defined on the Cartesian product of two ordinals.
A choice function is also called a transversal, but by transversalone frequently
means a 1-1 choice function. Given a set x of pairwise disjoint nonempty sets, a
set y E u x such that u n y consists of exactly one element for any u E x will also be
called a transversal (of x ) ; the logic behind this terminology is that y can be
identified with a 1-1 choice function for the identity function on x in a
straightforward manner. The reader will have no difficulty in finding out from
the context in which sense the word transversal is used.
The chapters in this book are numbered by Roman numerals; each chapter is
divided into sections, which are numbered by Arabic numerals independently of
chapter numbers. Theorems, lemmas, corollaries, definitions, and problems are
indexed by two numerals, the first of which is the section number and the second
is consecutively increasing within each chapter, and they are referred to by these
numbers. Some of the formulas in each section are indexed by a single number in
parentheses. In the same section these formulas are referred to by their numbers
in parentheses; when referring to a formula in a different section, we placed the
section number in front of the formula number. Occasionally we found it clearer
to divide a section into subsections. In this case, the subsections are numbered by
two numerals, and theorems, lemmas, corollaries, etc. are not numbered at all.
Each subsection contains at most one theorem, lemma, corollary, etc. (but it may
contain one of each); so it will not lead to misunderstanding if we refer to e.g. the
theorem in Subsection 4.1 as Theorem 4.1.
2. THE AXIOMS OF ZERMELO-FRAENKEL SET THEORY
Here we present a list of the axioms of Zermelo-Fraenkel set theory; the
axiom are written in the language of ZF, extended as described in the preceding
section.
2.1. The Axiom of the Empty Set:
3xVyl y E x .
This axiom simply says that there is a set with no elements (called the empty set,
and denoted by 0).
2.2. The Axiom oflExtensionality:
Vxy[Vz[z
Ex
oz
Ey
l o x =y ]
.
14
MTRODUCTlON CH.
I
Instead of the second equivalence sign c>, one can write * only, since the
implication G follows from an axiom of logic involving equality saying that
equal objects can replace each other in a formula.
2.3. The Axiom of Pairing:
Vxy3zVu[u E z o [ u = x v u = y ] ] .
This axiom together with the preceding one confirms the existence of an
unordered pair; namely, { x y ) is the unique z such that the part after the
quantifier 3z of the above formula holds. Singleton x can here be considered as
the unordered pair { x x ) .
2.4. The Axiom Scheme of Replacement: This axiom says that for every
operation F and for any set x , F"x is a set. As the quantifier 'for every operation F'
is inadmissible in the (extended) language of ZF (namely, quantification can be
made only over sets), this is actually not a single axiom, but infinitely many; that
is why it is called an axiom scheme. Formally, this scheme can be described as
follows: for every formula q ( x , y, 23 of ZF (here i stands for a finite sequence
etc.) we have
z,, . . ., z,- of variables; V i stands for Vz, . . . z,_
v.2 CVXYY' C(cp(x,y , 23
*Vu3uVy[y
Eu
d x , Y', 23)*Y = Y'l =
o 3 x E ucp(x,y , i ) ].]
It is easy to derive from this axiom scheme and from Axiom 2.1 the Axiom Scheme
of Comprehension, which says that the classifier { x E u : cp(x, ?); always defines a
set (we leave the formal description of this axiom scheme to the reader).
2.5. The Axiom of Union:
vx3yVtl[u E y 0 3 z C u E 2 & 2 E X I ] .
The unique set y satisfying this axiom is u x . Here, of course, one needs only
the.implication a, since the other implication can be made to hold with the
aid of (an instance of) the Axiom (Scheme) of Replacement (or, rather,
Comprehension). This economy is, however, usually only worth making if one
wants to prove that the axioms of set theory hold in a model.
2.6. The Axiom of the Power Set:
Vx3yVu[u c x e u E y ]
.
The unique set y satisfying this axiom is the power set of x, denoted by 9 " ~ ) .
THE AXIOMS OF ZERMELeFRAENKEL SET THEORY
15
2.7. The Axiom of Infinity:
3x[O E x & Vy[y
Ex*yu
( y ) E x]] .
This axiom confirms the existence of a set containing the sets 0, 1 =
= (0, {O; i, etc; the smallest such set is the ordinal w.
2=
2.8. The Axiom of Regularity:
Vx[x
# 0 3 3 y E x[x ny = 0]]
.
The Axiom of Regularity is convenient, but not necessary, for the development of
set theory. With the aid ofthe other axioms ofZFC (the Axiom ofchoice (AC)is
needed for this), it can be shown to be equivalent to the following statement:
there is no infinite sequence (xi:i <o)of sets such that . . . E x 2 E X , E xo holds.
It is important to see the difference between the Axiom of Regularity and wellfoundedness; this is discussed in Subsection 4.5 below.
The axioms mentioned so far constitute Z F set theory. If the next axiom is also
added, then we obtain the theory ZFC.
2.9. The Axiom of Choice (AC): Given a set x such that 0 $ x, call afunction
5 x- u x a choice function on x if f ( y )E y holds for every y E x (a choice function
f o r another function was defined above). The Axiom of Choice says that there is a
choice function on every set not containing the empty set.
2.10. This completes our list of the axioms of ZFC set theory. We shall assume
that the reader is familiar with the way to develop set theory from them, or from
other similar axioms (see e.g. Cohen [1966], Devlin 119731, Godel [1940], Jech
[1971], Kelley [1955], Takeuti-Zaring [1971], etc.) although usually a
familiarity with intuitive set theory also suffices. Yet to give some help to the
reader we shall prove some of the basic results of set theory in Section 4. Our
discussion cannot, however, be regarded as complete, since we shall accept some
simple facts of set theory as well known; in particular, after giving the definition
of ordinals in Section 3 we shall assume as known such standard results as e.g. the
fact that the ordinals are wellordered by the membership relation.
2.1 1. There are two more axioms that are occasionally mentioned in this book
when discussing consistency results. They are Godel’s Axiom of Constructibility,
usually abbreviated as V = L , and Marfin’s Axiom; the latter has a cardinal
parameter K and is abbreviated as MA,. The axiom V = L says that all sets are
constructible, but we cannot go into details as to what this means (see e.g. Godel
[1940], Cohen [1966], Devlin [1973], etc.); this axiom is well known to be
16
INTRODUCTION CH. 1
consistent with the axioms of ZFC provided ZF is consistent. The axiom MA, is
the following proposition : Given any notion of forcing c satisfying the countable
chain condition, for any set f of cardinality IK of dense open subsets of c there is
an figeneric filter on c. A notion of forcing c is a partially ordered set with a
largest element, usually denoted as 1. Two elements of c are compatible if they
have a common lower bound. c satisfies the countable chain condition if any
uncountable subset of c contains two distinct compatible elements. Denoting the
partial order on c by I, a subset x of c is den& open if Vp E c34 E X 4 5 p and
V p ~ c V q ~ x x C p s q = . p ~ x ] . Iisaset
ff
ofdenseopensubsetsofc,thenasetdccis
called an $generic filter if (i) 1 Ed, (ii) V p ~ d V Eq c C p ~ q = q~ d ] (iii)
,
Vp,qEd3r ~d[r~p,q],and(iv)Vx~f[xnd#O].)Asiswell known,ZFC+MA,
always implies that Po > K. The phrase ‘‘Po= I and Martin’s Axiom holds”
means that Po= I and MA, holdsfor all K < 1.It is well known that this assertion
is consistent relatively to ZFC whenever 1 > w is regular and is defined in a
‘simple’ way (e.g. I = K , , K,, K,, . . . ) (see e.g. Jech [1971], Martin-Solovay
[19721, and Solovay-Tennenbaum [ 19711).
3. ORDINALS, CARDINALS, AND ORDER TYPES
A set x is transitive if x c Y ( x ) ; it is an ordinal if it is transitive and all its
elements are also transitive; this definition is applicable only if the Axiom of
Regularity (see 2.8) is adopted, and in its absence another, more complicated
definition must be given. The above definition implies that, as mentioned in
Section 1, we accept J. von Neumann’s convention that an ordinal is the set of all
smaller ordinals. The class of all ordinals is often denoted by On. Greek lower
case letters other than E, 1, 0,u, cp, x and II/ (as for w, see below) will always denote
ordinals, unless the contrary is explicitly mentioned. Ordinals may, of course,
be denoted also by other letters. If a E B, then we may also write a < B (a is less
than, or precedes, etc., B); a s p means that a is less than or equal to 8, i.e., that
a </I
v a= B. For a set or class X of ordinals, U X and ( 7 X will usually be
denoted by sup X and min X, respectively; if X has a largest element (e.g. if it is
finite), then we may write max X instead of sup X; if X is empty, then min X is the
universal class according to thisdefinition, but it is better to think intuitively that
min X is not defined at all. The (ordinal)sum of the ordinals a and B is denoted by
a 4 8, and their (ordinal) product, by a . fl (the latter notation is ambiguous, see
below). We d o not define ordinal multiplication here, as the reader is probably
familiar with it (see e.g. Devlin [1973], Takeuti-Zaring [1971], etc.); we only
point out, for the reader’s orientation, that according to (the somewhat
arbitrary) tradition we have e.g. 2 w.= 2 42 4 2 4. . .=a, and w * 2 =
ORDINALS, CARDINALS, A N D ORDER TYPES
17
= o i o ( #a).
Given a sequence ( a v : v < q ) of ordinals, their sum is written as
For an ordinal a, u l 1 is the ordinal immediately preceding a if there is such
an ordinal; otherwise c1L 1 =a. Given an integer (i.e., a finite ordinal) r, we put
a L ( r / I ) = ( u l r ) L l and aLO=a. This defines u L r for all integers r. Let
a > O ;if u 2 1 = u, then c1 is called a limit ordinal, and otherwise a successor ordinal.
We shall occasionally be concerned with intervals of ordinals: [a, /?I=
= { {: a I { 5
[a, /?)= {<:a
I < /?I; (a, fl and (a, 8) are defined analogously.
Two sets x and y are equipollent if there is a 1 - 1 function .f from x onto y (i.e.,
such that f is 1 - 1 , dom ( f ) = x , and ra (f)=y).One often says equivalent instead
of equipollent, but the former term is clearly overused, and so we prefer to avoid
it. A cardinal is an ordinal that is not equipollent with any smaller ordinal; in this
way, as mentioned in Section 1 , we make no distinction between cardinals and
their initial ordinals. The lower case Greek letters K, I,p, t~ and t (with or without
subscripts or superscripts) always denote cardinals; ofcourse, cardinals may also
be denoted by other letters. The cardinality 1x1 of the set x is, by definition, the
(only) cardinal equipollent to x (the existence of such a cardinal follows from the
Wellordering Theorem - see Subsection 4.6). The (cardinal) sum of K and 1is
denoted by K + 1,and their (cardinal)product, by K .A.It might lead to confusion
that two very different concepts, ordinal product and cardinal product, are
denoted in the same way. In practice, if a and p are both cardinals, then a . /3 will
always denote the cardinal product when not mentioned otherwise explicitly;
and if a or p is not a cardinal, then a . /? cannot but denote the ordinal product.
Infinite cardinal sums will be denoted by the symbol C; e.g. if K , , x E I are
cardinals, then their sum is denoted by
K, or C ( K , : x E I ) . This sum isdefined
a),
<
K€I
as the cardinality of the set u { z , : x E I > , where the sets z, x E I, are pairwise
disjoint and such that (z,I = K , . One might be tempted to use also the notation
X{K,: X'E l ; ,but this ic incorrect, since e.g. if K , = K for all x E I, then {K,: x E I ) =
= { K ] , and so we should have C{K,: x E I ) = C{K > = K , and this is clearly not what
one wants. A similar remark applies to cardinal multplication, denoted by n.
The cardinal immediately succeeding K is denoted by K'. (Thus, e.g. for finite K
we have K + = K + 1 = K / 1.) K - denotes the cardinal immediately preceding K if
there is such a cardinal; otherwise we put K - = K . Let K > O ; if K - = K then K is
called a limit cardinal, and otherwise, a successor cardinal. Cardinal subtraction
can also be defined: if K and 1are cardinals, then K - I denotes the cardinality of
the set K \ 1.If K is infinite, then clearly K - 1= K or 0 according as I < K or A 2 K.
2 Combmatorial
18
INTRODUCTION CH.
1
Cardinal exponentiation is defined as follows: we put K'=I'KI, &'=
C
p',
P<K
K'=
1 K ~ and
,
P<'
K?=
1
p". One way of reading these symbols is, in
p< K . 0 <1
turn: " K to the power A", "weak K to the power A", " K to the weak power A", and
"weak K t o the weak power A". For a set x and a cardinal K we write [XI"=
= {y :y c x & Iy(= K ) and [XI<'= (y: y c x & lyl< K}. The symbols [XI'", [XI'",
and [x]"" may be defined analogously. It is well known that I9(x)l=2Ix1 holds
for any set x; and, moreover, for infinite x we also have I[x]"I=IxI' provided
~ 5 1 x (ifK>IxI,
1
thenI[x]"l=O),and I[x]<'I=lxl~provided K < I x I + .
Weshall
often use the symbol exp for iterated exponentation:
expo ( K ) = K
and
exp,+l(K)=expi (2"),
(1 1
where i is an integer; this defines exp, ( K ) for every integer i recursively. The
logarithm operation, discussed in detail in Section 7, will be defined as follows:
Here, in a natural way, Ais called the base of the logarithm. The iterated logarithm
(of base 3) is defined by recursion as follows:
L'(K)=K,
and
Lit ' ( K ) = L ~ ( L ~ ( K ) ) ,
(3
where i is an integer; this defines L i ( ~for
) every integer i.
For each ordinal a, K, denotes the (a i1)-st infinite cardinal; w , means the
same thing. The existence of these two different notations is due to the tradition
according to which K, denoted the cardinal and w, its initial ordinal; present day
axiomatic set theory, however, usually identifies the two, and, as mentioned
above, we also follow this practice. Nonetheless, occasionally this tradition may
be helpful, as we shall see this when discussing partition symbols in Subsection
8.2 (cf. the remarks made after (8.5)). One usually writes w instead of wo, but
neverKinsteadofK,.Let ~ = K , ; t h e n ~=' K a + l a n d ~ - = K , I laccordingtothe
notations introduced above; H a + , will also be denoted as IC(+@), and, for an
integer r, Ha:, as I&').
The Continuum Hypothesis, denoted as CH, is the assertion that 2'0=Ec1
holds. The Generalized Continuum Hypothesis, denoted as GCH, says that 2" =
= K + holds for any infinite cardinal K.A well-known result of Godel [1940] (see
also Cohen [1966], Devlin [1973], Jech [1971], etc.) says that GCH cannot be
disproved in ZFC, and an also well-known result of Cohen [1963] and [1964]
says that even CH cannot be proved in ZFC (see also e.g. Shoenfield [1971], Jech
[197 11).
19
ORDINALS, CARDINALS, AND ORDER TYPES
A partial order (x, <) is an ordered pair that consists of a set x and of a
relation < on x such that the relation < is irreflexive, antisymmetric and
transitive. A partial order (x, <) is called an order (occasionally: a total order or
a linear order) if it is trichotomous, i.e., if Vu, u E x [u<u v u = u v u<u]. The
relation 4 itself is called a (partial or total, according to which is the case)
ordering. A wellorder (x, <) is a total order such that each nonempty set y c x
has a <-least element (i.e., 3u E yVu E y[u<u], where, as usual, u<u stands for
u < u v u = u . A partial order is often called a partially ordered set, and the
analogous terms are also used for linear orders and wellorders.
Two partial orders (x, <) and (x', <') are called similar if they are
isomorphic as structures, i.e., if there is a 1-1 function f from x onto x' such that
Vu, u E x[u<v-f(u)<'f(u)].
It is well known that each wellorder is similar to an
ordinal a (more precisely to the wellorder (a, E lor), where la is the elementhood
relation restricted to a; this latter wellorder is usually written imprecisely as
denoted
,as
( a , € ) or (or, <)). The order type of a wellorder (x, i)
tp (x, <), is defined as this unique ordinal. The order type tp (x, <) of an
arbitrary partial order (x, <) can be defined in a more complicated way as
follows: If (x, <) is a partial order, then put
tp (x,
<) = {(lxl,
<I):
the partial order (1x1,
<') is similar to (x, <)};
so as not to run into contradiction with the definition of order type for
wellorders, t he latter definition must not be applied if (x, < ) is a wellorder. Since
the definition of the cardinality 1x1 of x as given hbove depends on the
Wellordering Theorem, i.e. on the Axiom of Choice, the definition of order type
given here also depends on this axiom. By formula (4.14) we shall give a possible
definition of cardinality that is independent of the Axiom of Choice. In an
entirely analogous way, the order type of a partially ordered set could also be
defined independently of this axiom.
Let d and d' be two order types. We write d I d ' if there are two partial orders
(x, <) and (x', <') such that tp(x, < ) = d , tp(x, < ) = d ' , x c x ' , and the
restriction <'lx of <' onto x is <. If d s d ' and d ' s d , then one writes d <d'. We
must emphasize that the relation Iis not an ordering even for linear order types
(i.e., for order types of linearly ordered sets). To give a simple example, denote by
q the order type of the set of rational numbers. Then q s q 1 sq, where +
denotes addition of order types. If d is an order type, then d* denotes the reverse
of d ; i.e., if d = t p (x, <), then d*=tp(x, >), where u>u means, as usual, the
same as v<u for any u, u E x.
+
20
INTRODUCTION CH.
Let (x,
and
I
<) be a partial order and d an order type. We put
[x, < I d = ( y : y E x & t p ( y , < I y ) = d ) ,
[x,
<]
-Cd
= ( y: y E x & tp
(y, <I y) < d j ;
can be defined analogously. When
the sets [x, <I“, [x, <Izd, and [x,
there is no danger of misunderstanding, we may write [xId, [x] <d, etc. instead of
the above symbols. (There might be a real danger of misunderstanding,
especially in cased is a cardinal, since then d is an order type as well according to
our terminology.)
If (x, <) and (x‘, <’) are total orders such that X ’ G X , <’=<lx’, and
is an initial segment of’
Vy E x‘Vz E X [z<y=z EX’] then we say that (x’,
(x, <);ifVy E X’VZ E X [y<z*z E x’] holds instead, then we say that (x’, <’) is
a final segment of (x, <).Alternatively, we may say that x‘ is an initial or ,final
segment, respectively, in (or, occasionally, of) (x, <).
Foratotalorder(x,<)andasety~x,yissaidtobecofinalin (x, < ) i f V u E X
3u E y[u<u]. If there is no danger of misunderstanding, then we may say that y is
cofinal in x. For an ordinal a, its cofinality, cf (a), is defined as
< I )
cf (a)=min (tp (x, <): x ~ and
a x is cofinal in (a, <)) .
(4)
Note here that tp(x, < ) on the right-hand side is an ordinal, and so the
minimum there is meaningful. If a and p are two ordinals, then the phrase ‘a is
cofinal to
means that cf ( a )= cf (p). Clearly cf ( a )= 0 implies a = 0, cf (a)= 1
holds exactly if a is a successor ordinal, and cf (a)? o holds otherwise. The basic
facts about the operation cf will be proved in Subsection 4.10, and in Subsection
4.1 1 this operation will be extended to order types.
Let K ? o be a cardinal. K is called regular if cf (K ) = K , singular if cf ( K ) < K ,
s t r o n g h i t if 2’< K for any A < K (in this case K - = K , i.e., K is a limit cardinal), and
inaccessible if it is an uncountable regular strong limit cardinal. An ordinal is
called accessible if it is not an inaccessible cardinal. A cardinal is said to be a
regufar fimit cardinal if it is a regular cardinal that is a limit cardinal; an
uncountable regular limit cardinal is occasionally called weakly inaccessible.
According to earlier terminology, a cardinal that we called inaccessible here was
called strongly inaccessible;we shall, however, never use the terms strongly and
weakly inaccessible.
21
BASIC TOOLS OF SET THEORY
4. BASIC TOOLS OF SET THEORY
We do not intend to present a full development of set theory from the axioms;
for example, we entirely skip the proofs of fundamental facts concerning ordinals
and cardinals. Our aim here is to give ‘axiomatic’ proofs of such basic results as
the Transfinite Recursion Theorem, the Wellordering Theorem, and Zorn’s
Lemma, etc, and to prove some fundamental facts about the cofinality operation. The reader need not be acquainted with such axiomatic proofs, since an
acquaintance with the development of intuitive set theory usually sufices for the
understanding of the material below; we felt, however, that our introductory
chapter would be incomplete without the inclusion of this section. E.g. the proof
of the Generalize2 Transfinite Recursion Theorem, given in Subsection 4.2, is
hard to find in textbooks on set theory.
4.1. TRANSFINITE
RECURS~ON
THEOREM.
For eaery operation G there is a unique
operation F on the class of all ordinals such that F ( a ) = G(F a ) holds .for all
ordinals a.
This formulation exploits the fact that G(x) was defined (as the empty set)even
in case the set x does not belong to the domain of G. Observe also that the above
theorem cannot be formalized in the language of ZF, as we cannot quantify over
classes because, by the definition of classes above, this would mean
quantification over formulas. Hence, this theorem is not a theorem of the formal
system of ZF (or ZFC); rather it is a theorem about that formal system. Such a
theorem is called rnerafheorcrn.Metatheorems in this section are Theorems 4.1,
4.2, Lemmas 4.1, 4.3, 4.4, and the R-Induction Principle in Subsection 4.4.
A
PROOF. Define F by the classifier
where a runs over ordinals. For every a there is at most one z such that ( a z ) E F ;
in fact if a is the least ordinal for which there are z and z’ with ( a z ) , (az’) E F and
z f z ’ , then there are functions f and f ’ such that dom ( f ) = d o m ( f ’ ) = a i 1,
f i ” a = f ‘ P a , z = f ( a ) = G ( f P a ) and z ’ = . f ‘ ( a ) = G ( , f ’ , ” a ) # z ,which is absurd.
For every ordinal a there is a z such that ( a z ) E F. In fact, if a is the least ordinal
for which there is no such z, then f ‘ = ( ( f l y ) E F : p < a ) is a set by the Axiom of
Replacement, and so it is obviously a function on a. Putting z = G ( f ’ ) ,it follows
from (1) withf=f’u((az)j that (az)EF. Hence F is indeed an operation on all
ordinals. The uniqueness of F also easily follows: If F and F‘ are two operations
22
MTRODUClTON CH. I
such that F(or)= G ( F Pa) and F'(a)= G(F' (*a) for all ordinals c1, then we get a
contradiction for the first a for which F ( a ) # F ' ( a ) . The proof is complete.
4.2. RECURSION
ON WELL-FOUNDED RELATIONS. A relation R on a class X is wellfounded if (y E X: y R x ) is a set for every x E X and
Vu E X [u # O=dx
E uVy E u
1y R x ] .
(2)
The element x whose existence is claimed in this formula is called R-minimal in u.
From the Axiom of Choice it easily follows (eg. with the aid of the preceding
theorem) that the second requirement (in the presence of the first one) is
equivalent to saying that there is no infinite sequevce ( x i :i < o)ofelements of X
such that x i + Rxi holds for any i <a.
GENERALIZED
TRANSFINITE
RECURSION
THEOREM.
l f R is a well-founded relation on
a class X then for every operation G there is a unique operation F on X such that
F(x)= G(F ( yE X : ~ R x ) . )
(3)
holds for every x E X .
Usually, this result is proved as a corollary of the preceding theorem. Such a
proof is intuitively quite clear and proceeds as follows. First one defines an
R-rank, that is, an operation rk, on X into the ordinals with the following property: if xRy, then rk,(x) < rk,(y) holds; having done this, the definition of F can
be given with the aid of common transfinite recursion (i.e., with the aid of the
preceding theorem). Of course, one has to define the operation rk, also by
common transfinite recursion, which is a little complicated; its definition by
generalized transfinite recursion (i.e., with the aid of the present theorem) is rather
simple: we put
u
(4 1
rk, ( X ) = [rk, b)41: YRX)
for any x E R.
As the proof outlined above is formally quite complicated, we shall rather give
a direct proof analogous to the proof of the preceding theorem. The advantage of
this proof is that it immediately gives the definition of the operation F. To this
end, for a relation R on a class X, call a class D G X R-transitive if
Vx E DVy[yRx*y E 01.
We need the following lemma (this lemma is needed also in the alternative proof
outlined above):
TRANSITIVE
CLOSURE
LEMMA.
If a relation R on a class X is such that the class
{ y E X : y R x ) is a set for every x E X , thenfor every set u E X there is an R-transitive
set d including u.
23
BASlC TOOLS OF SET THEORY
We shall use this lemma only in case R is a well-founded relation; such a
relation satisfies the requirements of the lemma by the definition of wellroundedness.
PROOF
OF THE LEMMA. Given R, X and U E X , define the operation F by
transfinite recursion (i-e.,by Theorem 4.1) as follows:
F(a)= u u { y :3fl< a32 E F ( p ) b = z v y R z ] ; .
It is very important here that F ( a ) is always a set, and never a real class; the
reason why this is so important can be seen from the formal definition of F given
according to Theorem 4.1: For any set U EOn x X put
G(v)=(yE X :y
Eu
v 3az[(az)
E u&
( y = z v y R z ) ] ].
G(v) is Q ser for every u by the assumptions of the lemma to be proved; hence we
can define an operation G a$
G = { ( u G ( u ) ) :u%On x XI.
We can now define the operation F by the stipulation that F ( a )= G(F P a ) for
every a, as is done in Theorem 4.1.
Write now
d=
u
F(n).
n<w
It is easy to see that d is R-transitive, since if x ~d then x E F ( n ) for some n KO,
and then y E F(n 4 1 ) for any y E X with y R x . On the other hand, U E F(a) clearly
holds for every ordinal a, and so u c d ; hence d is an R-transitive set including u
(actually, as is easily seen, it is the smallest such set; hence it might be called the
R-transitive closure of u). The proof is complete.
PROOF OF THE THEOREM. Given R, X and G , write analogously to (1) that
F = { (xz): x E X & 3fd[f
& d=dom(f)
is a function &
& d c X is R-transitive & x ~d
&
& V U E ~ [ ~ ( U ) = G ( ~ / \ { ~ E X :&~ Rz U
= ~) () x] ) ] ) .
We repeat the proof of Theorem 4.1 with minor changes.
First we claim that, for every x, there is at most one z such that (xz) E F. In
fact, assume that (xz,), (xz,) E F and z o # z l ; then there are fo, do and f,,d ,
such that the formula after the existential quantifier in ( 5 ) holds with z, f,, do
andz,, f,,d, replacingz,f,d,respectively.Writed‘=d,nd,; thend’isobviously
R-transitive. Let x’ be an R-minimal element in the set ( y E d : fo(y)#fl(y)} (see
24
INTRODUCTION c n .
I
the definition right after formula (2)). Note that this set is not empty since x
belongs to it, and so there is such an x’ by well-roundedness. Then ,fo(x‘)# f,(x’),
and yet fo(y)= f,(y) for every y in the set {y E d’: yRx’).= {y E X: yRx’). (this
latter equality holds by the R-transitivity of d‘).This is a contradiction, since we
have
. f b ( x ’ ) = G ( f O , ~ { y ~ X : y R ~ ’ ) ) =’{yEX:yRx’).)=f,(x’).
G(f,
Hence our first claim is proved.
Secondly, we claim that for every x E X there is a z such that (xz) & F. In fact,
assume x, E X is such that (xoz) E F holds for no set z, and let c be an Rtransitive set containing x,; there is such a c by the lemma just proved.
Put
d’ = (XE C: 3z( XZ} E F). .
We assert that d’ is an R-transitive set. In fact, if x E d ’ , then (xz) E F for some z,
and so the formula after the existential quantifier in (5) is satisfied with some f
and d. If now yRx holds then y E d by the R-transitivity of d, and so (yf(y)) E F
holds by (5). As y E c by the R-transitivity of c, we can conclude that y E d‘. This
proves our assertion that d’ is R-transitive.
Write
f ” = ((yz} E F: y Ed’).
Since for every y E X there is at most one z such that (yz) E F according to our
first claim, f ‘is a set (and so a function) by the Axiom of Replacement. dom (f‘)=
=d’ holds by the definition of d’. Let now x be an R-minimal element in c\d‘; as
x, E c\d’ by the definitions of c, xo, and d’, c\d’ is not empty, and so there is such
an x by (2). Noting that
{y~X:yRx)cd’
(6)
holds by the R-minimality of x and by the R-transitivity of d‘, put
z = G ( f r{y E X: ~ R x ) . ) ,
and
Noting that d is R-transitive by (6) since d’ is so, we can see that x, z, f a n d d
satisfy the formula after the existential quantifier in ( 5 ) ; hence (xz) E F. This is a
contradiction, proving our second claim that Vx E X ~ Z ( X ZE )F.
Finally, we claim that there is only one operation F satisfying (3). Assume, on
the contrary, that F and F’ are two such operations with F(xo)# F‘(xo)for some
xo E X. Let d s X be an R-transitive set containing xo (there is such a d by the