Achim Blumensath blumensath@mathematik.tu-darmstadt.de is ...

Achim Blumensath
blumensath@mathematik.tu-darmstadt.de
�is documentwas lastupdated2012-08-08.
�e latestversion can be found at
www.mathematik.tu-darmstadt.de/~blumensath
Copyright2012 Achim Blumensath
Allrights arereserved.Permission is grantedto distribute and copythis
documentunderthe followingterms:
◆ �euse isprivate and non-commercial. Inparticular,youare not
allowed tosellprinted copies ofthis document.
◆ All changes,additions,and omissionstothe document are clearly
marked assuch.
◆ All excerpts, quotes, and translations contain an acknowledgement
ofthe original.

Contents
A. Set�eory 1
a1 Basic set theory 3
1 Sets and classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Stages and histories . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 �e cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . 18
a2 Relations 27
1 Relations and functions . . . . . . . . . . . . . . . . . . . . . . . 27
2 Products andunions . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Graphs andpartial orders . . . . . . . . . . . . . . . . . . . . . 39
4 Fixedpoints and closure operators . . . . . . . . . . . . . . . . 47
a3 Ordinals 57
1 Well-orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2 Ordinals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3 Induction and fixed points . . . . . . . . . . . . . . . . . . . . . 74
4 Ordinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 85
a4 Zermelo-Fraenkel set theory 105
1 �e Axiom of Choice . . . . . . . . . . . . . . . . . . . . . . . . 105
2 Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3 Cardinal arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 116
4 Cofinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
logic, algebra & geometry2012-08-08 — ©achim blumensath v

Contents
5 �e Axiom ofReplacement. . . . . . . . . . . . . . . . . . . . . 131
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
B. Universal Algebra 137
b1 Structures and homomorphisms 139
1 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 146
3 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4 Congruences andquotients . . . . . . . . . . . . . . . . . . . . 160
b2 Trees and lattices 173
1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3 Ideals and filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4 Prime ideals andultrafilters . . . . . . . . . . . . . . . . . . . . 194
5 Atomic lattices andpartition rank . . . . . . . . . . . . . . . . . 200
b3 Algebraicconstructions 213
1 Terms andterm algebras . . . . . . . . . . . . . . . . . . . . . . 213
2 Direct andreducedproducts . . . . . . . . . . . . . . . . . . . . 224
3 Direct and inverse limits . . . . . . . . . . . . . . . . . . . . . . 232
b4 Topology 243
1 Open and closedsets . . . . . . . . . . . . . . . . . . . . . . . . 243
2 Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . 248
3 Hausdorffspaces and compactness . . . . . . . . . . . . . . . . 252
4 �eProducttopology . . . . . . . . . . . . . . . . . . . . . . . . 259
5 Densesets and isolatedpoints . . . . . . . . . . . . . . . . . . . 263
6 Spectra andStone duality . . . . . . . . . . . . . . . . . . . . . . 272
7 Stonespaces and Cantor-Bendixsonrank . . . . . . . . . . . . 279
vi

Contents
b5 ClassicalAlgebra 287
1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
2 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
3 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
5 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
6 Ordered fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
C. First-Order Logic and its Extensions 343
c1 First-order logic 345
1 Infinitary first-order logic . . . . . . . . . . . . . . . . . . . . . 345
2 Axiomatisations . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
3 �eories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
4 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
5 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
6 Extensions of first-order logic . . . . . . . . . . . . . . . . . . . 383
c2 Elementarysubstructures andembeddings 395
1 Homomorphisms and embeddings . . . . . . . . . . . . . . . . 395
2 Elementary embeddings . . . . . . . . . . . . . . . . . . . . . . 400
3 �e�eorem of Löwenheim andSkolem . . . . . . . . . . . . . 405
4 �e Compactness �eorem . . . . . . . . . . . . . . . . . . . . 412
5 Amalgamation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
c3 Typesandtypespaces 427
1 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
2 Typespaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
3 Retracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
4 Localtypespaces . . . . . . . . . . . . . . . . . . . . . . . . . . 456
5 Stabletheories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
vii

Contents
c4 Back-and-forthequivalence 473
1 Partial isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 473
2 Hintikka formulae . . . . . . . . . . . . . . . . . . . . . . . . . . 482
3 Ehrenfeucht-Fraïssé games . . . . . . . . . . . . . . . . . . . . . 485
4 κ-complete back-and-forthsystems . . . . . . . . . . . . . . . . 494
5 �etheorems of Hanf and Gaifman . . . . . . . . . . . . . . . . 501
c5 General modeltheory 509
1 Classifying logical systems . . . . . . . . . . . . . . . . . . . . . 509
2 Hanf and Löwenheim numbers . . . . . . . . . . . . . . . . . . 513
3 �e�eorem of Lindström . . . . . . . . . . . . . . . . . . . . . 520
4 Projective classes. . . . . . . . . . . . . . . . . . . . . . . . . . . 532
5 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
6 Fixed-point logics . . . . . . . . . . . . . . . . . . . . . . . . . . 553
D. Axiomatisation and Definability 581
d1 Quantifierelimination 583
1 Preservationtheorems . . . . . . . . . . . . . . . . . . . . . . . 583
2 Quantifier elimination . . . . . . . . . . . . . . . . . . . . . . . 587
3 Existentially closedstructures . . . . . . . . . . . . . . . . . . . 597
4 Abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
5 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
d2 Productsandvarieties 615
1 Ultraproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
2 �etheorem of Keisler andShelah . . . . . . . . . . . . . . . . 620
3 Reducedproducts and Horn formulae . . . . . . . . . . . . . . 633
4 Quasivarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
5 �e�eorem of Feferman and Vaught . . . . . . . . . . . . . . 650
viii

Contents
d3 O-minimalstructures 655
1 Orderedtopologicalstructures . . . . . . . . . . . . . . . . . . 655
2 O-minimal groups andrings . . . . . . . . . . . . . . . . . . . . 661
3 Cell decompositions . . . . . . . . . . . . . . . . . . . . . . . . . 663
E. Classical Model �eory 683
e1 Saturation 685
1 Homogeneousstructures . . . . . . . . . . . . . . . . . . . . . . 685
2 Saturatedstructures . . . . . . . . . . . . . . . . . . . . . . . . . 690
3 Projectively saturatedstructures . . . . . . . . . . . . . . . . . . 701
4 Pseudo-saturatedstructures . . . . . . . . . . . . . . . . . . . . 705
5 Definability inprojectively saturated models . . . . . . . . . . 712
e2 Prime models 721
1 Isolatedtypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721
2 �e Omitting Types�eorem . . . . . . . . . . . . . . . . . . . 723
3 Prime and atomic models . . . . . . . . . . . . . . . . . . . . . 731
4 Constructed models . . . . . . . . . . . . . . . . . . . . . . . . . 735
e3ℵ0-categoricaltheories 743
1 ℵ0-categorical theories and automorphisms . . . . . . . . . . . 743
2 Fraïssé limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759
3 Zero-one laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765
e4 Indiscerniblesequences 773
1 Ramsey�eory . . . . . . . . . . . . . . . . . . . . . . . . . . . 773
2 Ramsey�eory fortrees . . . . . . . . . . . . . . . . . . . . . . 777
3 Indiscerniblesequences . . . . . . . . . . . . . . . . . . . . . . . 789
4 �e independence andstrict orderproperties . . . . . . . . . . 799
ix

Contents
e5 Functorsandembeddings 809
1 Local functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809
2 Word constructions . . . . . . . . . . . . . . . . . . . . . . . . . 816
3 Ehrenfeucht-Mostowski models . . . . . . . . . . . . . . . . . . 825
e6 Abstract elementaryclasses 839
1 Abstract elementary classes . . . . . . . . . . . . . . . . . . . . 839
2 Amalgamation andsaturation . . . . . . . . . . . . . . . . . . . 848
3 Limits of chains . . . . . . . . . . . . . . . . . . . . . . . . . . . 861
4 Categoricity andstability . . . . . . . . . . . . . . . . . . . . . . 865
F. Geometric Model �eory 873
f1 Geometries 875
1 Dependencerelations . . . . . . . . . . . . . . . . . . . . . . . . 875
2 Matroids and geometries . . . . . . . . . . . . . . . . . . . . . . 880
3 Modular geometries . . . . . . . . . . . . . . . . . . . . . . . . . 887
4 Strongly minimalsets . . . . . . . . . . . . . . . . . . . . . . . . 893
5 Vaughtian pairs andthe�eorem of Morley . . . . . . . . . . . 901
f2 Ranksand forking 913
1 Morleyrank and ∆-rank . . . . . . . . . . . . . . . . . . . . . . 913
2 Independencerelations . . . . . . . . . . . . . . . . . . . . . . . 927
3 Preforkingrelations . . . . . . . . . . . . . . . . . . . . . . . . . 940
4 Forkingrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . 953
f3 Simpletheories 965
1 Dividing and forking . . . . . . . . . . . . . . . . . . . . . . . . 965
2 Simpletheories andthetreeproperty . . . . . . . . . . . . . . . 975
3 �e Independence�eorem . . . . . . . . . . . . . . . . . . . . 992
x

Contents
Literature 1007
Symbol Index 1009
Index 1019
xi

PartA.
Set�eory

a1. Basicsettheory
1. Setsandclasses
In mathematicsthere are basically two waysto definethe objectsunder
consideration. Onthe one hand, one can explicitly constructthem from
already known objects. For instance, therational numbers andthereal
numbers areusually introduced inthis way. Onthe other hand, one can
takethe axiomatic approach,that is, one compiles a list of desiredproperties
and one investigates any object meetingtheserequirements.Some
well known examples are groups, fields, vector spaces, and topological
spaces.
Since set theory is meant as foundation of mathematics there are
no more basic objects available in terms of which we could definesets.
�erefore, we will followthe axiomatic approach. We will present a list
ofsix axioms and any objectsatisfying all ofthem will be called a model
of set theory. Such a model consists of two parts: (1) a collection S of
objectsthat we will callsets, and(2)some method which, giventwosets
a and b,tellsus whether a isanelement of b.
We will not care what exactlythe objects inSare or howthis method
looks like. For example, one could imagine a model of set theory consisting
of natural numbers. If we define that a natural number a is an
element of the natural number b if and only if the a-th bit in the binary
encoding ofbis 1,then all but one of our axioms will besatisfied. It
is conceivable that asimilar but more involved definition might yield a
modelthatsatisfies all ofthem.
We will introduce our axioms in a stepwise fashion during the following
sections. To help readers trying to look up a certain axiom we
logic, algebra & geometry2012-08-08 — ©achim blumensath 3

a1. Basicsettheory
include a complete list below even if most ofthe needed definitions are
still missing.
Axiom ofExtensionality. Two sets a and b are equal if, and only if, we
have x∈ a⇔x∈ b, for allsets x.
Axiom ofSeparation. If a is aset and φ apropertythen{x∈a∣φ} is
aset.
Axiom ofCreation. For everyset athere is asetS suchthat S is astage
and a∈S.
Axiom of Infinity. �ere exists asetthat is a limitstage.
Axiom ofChoice. For everyset Athere exists a well-order R over A.
Axiom ofReplacement. If F is a function and dom F is a set then so is
rng F.
Asking whether these axioms are true does make as much sense as
the question of whether the field axioms are true, or those of a vector
space. Instead, what we are concerned with istheirconsistency andcompleteness.
�at is, there should exist at least one object satisfying these
axioms and all such objects should look alike. Unfortunately, one can
provethatthere is no complete axiomsystem forsettheory. Hence, we
will haveto deal withthe factthatthere are many different models ofset
theory andthere is no wayto choose one ofthem asthe‘canonical one’.
Inparticular, there is nosuchthing as‘thereal model ofsettheory’.
Moreseriously, it is even impossibletoprovethat our axiom system
is consistent. �at is, it might be the case that there is no model of set
theory and we have wasted ourtimestudying a nonsensical theory.
�e firstproblem is dealt withrather easily. It does not matter which
ofthese models we are givensince anytheoremthat we can derive from
the axioms holds in every model. Butthesecondproblem isserious. All
we can do is to restrict ourselves to as few axioms as possible and to
hopethat no one will ever be ableto derive a contradiction. Of course,
the weaker the axioms the more different models we might get andthe
fewertheorems we will be abletoprove.
4

1. Setsandclasses
Inthe following we will assumethatSis an arbitrary but fixed model
of set theory. �at is, S is a collection of objects that satisfies all the
axioms we will introduce below. S will be called the universe and its
elements are calledsets. NotethatSitself is not asetsince we willprove
belowthat noset is an element of itself. By convention, if below wesay
thatsomesetexiststhen we meanthat it is contained inS.Similarly, we
saythatallsets havesomeproperty if all elements ofSdoso.
Intuitively, aset is a collection of objects called itselements. Ifa andb
aresets, i.e., elements ofS, we write a∈bif a isanelement of b and we
define
a⊆b : iff every element x∈a is also an element x∈ b.
If a⊆b, we call a a subset of b, and we say that a is included in b, or
that b is a superset of a. We use the usual abbreviations such as a⊂b
for a⊆band a≠b; a∋bforb∈a; and a∉bif a∈bdoes not hold.
Since aset is a collection of objects it is naturaltorequirethat aset is
uniquely determined by its elements. Our first axiom can therefore be
regarded asthe definition of aset.
Axiom of Extensionality. Twosets aandbareequal if,and only if,
x∈a iff x∈ b , forallsets x.
Lemma1.1. Twosets aandbareequal ifand only if a⊆bandb⊆a.
In orderto define aset we havetosay what its elements are. Iftheset
is finite we can just enumerate them. Otherwise, we have to find some
propertyφsuchthat an object x is an element of a if, and only if, it has
theproperty φ.
Definition1.2. (a) Let φ be aproperty.{x∣φ} denotestheset a such
that, for allsets x, we have
x∈a iff x haspropertyφ.
5

a1. Basicsettheory
If S does not contain such an object then the expression{x ∣ φ} is
undefined.
(b) Letb0,... ,bn−1 besets. We define
{b0,... ,bn−1}∶={x∣x=bi forsome i

already been defined,the nextstage
HFn+1∶={x∣ x⊆ HFn}
1. Setsandclasses
consists of allsetsthat we can construct from elements of HFn.
Aset is called hereditary finite if it is an element ofsome HFn.�eset
of all hereditary finitesets is
HF∶={x∣x∈ HFn forsome n}.
Notethat we cannotprove atthe momentthatHFreally is aset.Since
the emptyuniverseS=∅trivially satisfies the Axiom of Extensionality,
we even cannot show that the empty set exists without additional axioms.
Letus assume forthe moment that HF does exists. Its firststages
are
HF0=∅
HF1={∅}
HF2={∅,{∅}}
HF3={∅,{∅},{{∅}},{∅,{∅}}}
...
By induction on n, one can prove that HFn⊆ HFn+1 and each set a∈
HFn+1 is of the form a = {b0,...,bk−1}, for finitely many elements
b0,...,bk−1∈ HFn. Note that each stage HFn is hereditary finite since
HFn∈ HFn+1⊆ HF, buttheirunion HF is not because HF∉HF.
Exercise 1.1. Prove the following statements by induction on n. (Although
we have not defined the natural numbers yet, you may assume
for this exercise that they are available and that their usual properties
hold.)
(a) HFn⊆ HFn+1.
(b) HFn has finitely many elements.
7

a1. Basicsettheory
(c) Every set a∈HFn+1 is ofthe form a={b0,... ,bk−1}, for finitely
many elements b0,... ,bk−1∈ HFn.
HF can beregarded as an approximationtothe class of allsets. In fact,
all but one of the usual axioms of set theory hold for HF. �e only exception
isthe Axiom of Infinity whichstatesthatthere exists an infinite
set.
We can encode natural numbers byspecial hereditary finitesets.
Definition1.4. To each natural number n we associatetheset
[n]∶={[0],...,[n− 1]}.
�eset of all natural numbers is
N∶={[n]∣n a natural number}.
Notethat[n]∈HFn+1 but[n]∉HFn, and N∉HF. �is construction
can beusedto definethe natural numbers inpurelysettheoretic terms.
In the following by a natural number we will always mean a set of the
form[n].
It would be nice if there were a universe S that contains all sets of
the form{x∣φ}. Unfortunately, such a universe does not exists, that
is, if we add the axiom that claims that{x ∣ φ} is defined for all φ,
we obtain a theory that is inconsistent, i.e., it contradicts itself. In fact,
we can even showthatthere areproperties φ suchthat no model ofset
theory contains a set of the form{x∣φ}. And we can do so without
using asingle axiom ofsettheory.
�eorem1.5(Zermelo-RussellParadox). {x∣x∉ x} is notaset.
Proof. Supposethattheseta∶={x∣x∉x}exists. Letx be an arbitrary
set. By definition, we have x∈a if and only if x∉x. In particular, for
x=a,we obtain a∈a iff a∉a. A contradiction. ◻
To betterunderstand what is going on, letusseewhat happens ifwe
restrict ourselves to hereditary finite sets. �e set{x∈ HF∣x∉x}
8

1. Setsandclasses
equals HF since no hereditary finite set contains itself. But HF∉HF is
not hereditary finite.�esame happens inrealsettheory.�e condition
x∉ x issatisfied by all sets andwe have{x∣x∉ x}=S,which is not a
set.
In general, an expression of the form{x∣φ} denotes a collection
X⊆ S that may or may not be a set, i.e., an element X∈S. We will
call objects ofthe form{x∣φ}classes. Classes that are notsetswill be
called proper classes. If X={x∣φ} and Y={x∣ ψ} are classes and
a is aset,wewrite
a∈ X : iff a haspropertyφ ,
X⊆ Y : iff every setwithproperty φ also haspropertyψ ,
and X= Y : iff X⊆Y andY⊆ X.
If X is a proper class then we define X∉ Y, for every Y. Note that, if
X and Y are sets then these definitions coincide with the ones above.
Finally, we remark that every set a is a class since we can write a as
{x∣ x∈a}.
When defining classeswe haveto be a bit careful aboutwhatwe call
a property. Let us define a property to be a statement that is build up
from basicpropositions ofthe form x∈y and x=y by
◆ logical conjunctions like‘and’,‘or’,‘not’,‘if-then’;
◆ constructs of the form ‘there exists a set x suchthat ...’ and ‘for
allsets x it holdsthat...’.
(Such statements will be defined in a more formal way in Chapter c1
wherewewill callthem‘first-order formulae’.)�ingswe are not allowed
to say include statements of the form ‘�ere exists a property φ such
that...’ or‘For all classes X it holdsthat...’.
We have defined a class to be an object of the form{x∣φ} where
φ is astatement aboutsets.What happens ifwe allowstatements about
arbitrary classes? Note that, if φ is a property referring to a class X=
{x ∣ ψ} then we can transform φ into an equivalent statement only
talking aboutsets byreplacing allpropositions y∈ X, X∈y, X=y, etc.
bytheirrespective definitions.
9

a1. Basicsettheory
Example. Let X={x∣∅∉ x}.We canwritethe class
inthe form
{y∣ y≠∅and y⊆ X}
{y∣ y≠∅and∅∉ x for all x∈y}.
�esituation is analogoustothe case ofthe complex numberswhich
are obtained fromthereal numbers by adding imaginary elements.We
can translate any statement about complex numbers x+iy into one
about pairs⟨x,y⟩ of real numbers. Consequently, it does not matter
whetherwe allow classes inthe definition of other classes.
Intuitively, the reason for a proper class such as S not being a set is
that it is too ‘large’. For instance, when considering HF we see that a
set a⊆HF is hereditary finite if, and only if, it has only finitely many
elements. Hence, if we can show that a class X={x∣φ} is ‘small’, it
should form aset.What dowe mean by‘small’? Clearly, wewould like
every set to be small. Furthermore, it is natural to require that, if Y is
small and X⊆ Y then X is alsosmall. �erefore,we define a class X to
besmall if it is asubclass X⊆aofsomeset a.
Definition1.6. For a class A and aproperty φwe define
{x∈ A∣ φ}∶={x∣x∈A and x haspropertyφ}.
�is definition ensures that every class of the form X={x∈a∣φ}
whereais aset issmall. Conversely, if X={x∣ φ} issmallthen X⊆ a,
forsomeset a, andwe have X={x∈a∣φ}. Oursecond axiomstates
that everysmall class is aset.
Axiom of Separation. If a isasetandφaproperty thentheclass
isaset.
10
{x∈ a∣φ}

2. Stagesand histories
With this axiom westill cannot provethat there is any set. But if we
have at least oneset a,we can deduce, for instance, that alsothe empty
set∅={x∈ a∣x≠x} exists.
Definition1.7. LetA and B be classes.
(a) �e intersection ofA isthe class
⋂A∶={x∣ x∈y for all y∈A}.
(b) �e intersection ofA and B is
A∩B∶={x∣x∈A and x∈B}.
(c) �edifference between A and B is
A∖B∶={x∈ A∣ x∉ B}.
Lemma 1.8. Let a be aset and B aclass.�en a∩Band a∖B are sets.
If Bcontainsat least oneelementthen⋂B isaset.
Proof. �e factthat a∩B={x∈a∣x∈B} and a∖B aresets follows
immediately from the Axiom of Separation. If B contains at least one
element c∈Bthenwe canwrite
⋂B={x∈ c∣x∈y for all y∈B}. ◻
Notethat⋂∅=Sis not aset.
2. Stagesand histories
�e construction ofHF above can be extendedto one ofthe classSof all
sets.We defineSastheunion of an increasingsequence ofsetsSα , called
thestages ofS. Again,westartwiththe emptysetS0∶=∅. IfSα is defined
thenthe nextstage Sα+1 contains all subsets of Sα. Butthistime,we do
not stop when we have defined Sα for all natural numbers α. Instead,
11

a1. Basicsettheory
every time we have defined an infinite sequence of stages we continue
bytakingtheirunionto formthe nextstage.So oursequencestartswith
S0= HF0 , S1= HF1 , S2= HF2 , ...
�e nextstage a�er allthe finite ones isSω∶= HF andwe continue with
Sω+1={x∣ x⊆ HF} , Sω+2={x∣x⊆Sω+1} , ...
A�erwe have defined Sω+n for all natural numbers n we againtakethe
union
Sω+ω={x∣ x∈ Sω+n forsome n},
andso on.
Unfortunately, makingthis constructionpreciseturns outto bequite
technical since we cannot definethe numbers α yet thatwe need to indexthesequenceSα.�is
hastowaituntilSection a3.2. Instead,we start
by giving a condition for some set S to be a stage, i.e., one of the Sα. If
we order allsuchsets by inclusionthenwe obtainthe desiredsequence
S0⊆ S1⊆⋅⋅⋅⊆Sω⊆ Sω+1⊆⋯ ,
withoutthe needtoreferto its indices.
First,we isolatesome characteristic properties ofthesets HFn which
we would like that our stages Sα share. Note that, at the moment, we
cannotprovethat any ofthesets mentioned below actually exists.
Definition 2.1. LetA be a class.
12
(a) We call Atransitive if x∈y∈A impliesx∈A.
(b) We call A hereditary if x⊆y∈A impliesx∈ A.
(c) �eaccumulation of A isthe class
acc(A)∶={x∣there issome y∈Asuchthat x∈y or x⊆y}.
Notethat eachstage HFn of HF is hereditary andtransitive.

2. Stagesand histories
Exercise 2.1. By induction on n,showthattheset[n] istransitive. Give
an example of a number n suchthat[n] is not hereditary.
�e next lemmas follow immediately fromthe definitions.
Lemma 2.2. Let Abeaclass,andb,c sets. �e followingstatements are
equivalent:
(a) c∈b∈A implies c∈A,that is,A istransitive.
(b) b∈A impliesb⊆A.
(c) b∈A impliesb∩A= b.
Lemma 2.3. Let AandBbeclasses.
(a) A⊆ acc(A)
(b) IfBis hereditaryandtransitiveand ifA⊆ B,then acc(A)⊆ B.
(c) A is hereditaryandtransitive if,and only if, acc(A)= A.
Lemma 2.4. IfAandBaretransitiveclassesthenso is A∩B.
Exercise 2.2. Prove Lemmas2.2, 2.3, and 2.4.
Definition 2.5. LetA be a class.
(a) A minimalelement of A is an element b∈Asuchthat b∩A=∅,
that is,there is no element c∈Awith c∈b.
(b) Aset a is founded if everysetb∋a has a minimal element.
(c) �e foundedpart of A istheset
fnd(A)∶={x∈ A∣ x is founded}.
Example. �e emptyset∅andtheset{∅} are founded.Toseethat{∅}
is founded, consider asetb∋{∅}. If{∅} is not a minimal element ofb,
then b∩{∅}≠∅. Hence,∅∈ b is a minimal element ofb.
Exercise 2.3. Provethat every hereditary finiteset is founded.
13

a1. Basicsettheory
Wewill introduce an axiom belowwhich impliesthat every class has a
minimal element. Hence, everyset is founded andwe have fnd(A)= A,
for all classesA. Althoughthe notions of a foundedset andthe founded
part of asetwillturn outto betrivial,westill needthemto definestages
andto formulatethe axiom.
Lemma 2.6. IfBisahereditaryclassand a∈Bthen fnd(a)∈fnd(B).
Proof. For a contradiction suppose that fnd(a)∉fnd(B). Since B is
hereditary and fnd(a)⊆ a∈B, we have fnd(a)∈ B. Consequently,
fnd(a)∉ fnd(B) impliesthatthere issomesetx∋ fnd(a)without minimal
element. Inparticular, fnd(a) is not a minimal element of x,that
is,there existssomeset y∈x∩ fnd(a). But y∈ fnd(a) impliesthat y is
founded.�erefore, from y∈xit followsthatx has a minimal element.
A contradiction. ◻
Inthe language ofSection a3.1 the next theorem states that the membership
relation∈iswell-founded on every class oftransitive, hereditary
sets.
�eorem 2.7. Let Abeanonemptyclass. Ifeveryelementx∈ A is hereditaryandtransitive,then
A hasaminimalelement.
Proof. Choose an arbitrary element c∈A andset
b∶={ fnd(x)∣ x∈ c∩A}.
If b=∅ then c∩A=∅ and c is a minimal element of A. �erefore,
we may assume that b≠∅. Since c∈A is hereditary, it follows from
Lemma2.6thatb⊆fnd(c). Fixsomex∈ b⊆fnd(c).�enx is founded
andx∈ b impliesthatbhas a minimal element y. By definition ofb,we
have y=fnd(z), forsomez∈c∩A.
We claimthatzis a minimal element of A.Suppose otherwise.�en
there existssome element u∈z∩A.Since c istransitive we have u∈c.
Hence, u∈c∩Aimplies fnd(u)∈b. On the other hand, since z∈A
is hereditary it follows from Lemma 2.6 that fnd(u)∈fnd(z). Hence,
14

2. Stagesand histories
fnd(u)∈fnd(z)∩b≠∅ and y= fnd(z) is not a minimal element ofb.
A contradiction. ◻
We would like to define that a set S is a stage if it is hereditary and
transitive. Unfortunately, this definition is too weak to show that the
stages can be arranged in an increasingsequenceS0⊆ S1⊆⋯⊆Sα⊆⋯.
�erefore, our definitionwill beslightly more involved.To eachstageSα
wewill associate its history
H(Sα)={Sβ∣ β

a1. Basicsettheory
Proof. (a) a⊆a∈Himplies a∈ acc(H)= S.
(b) By definition of a history, we have a = acc(H∩a). Hence, if
we can show that H∩ais a history then its stage is a. Clearly, every
element of H∩a⊆His hereditary andtransitive. Letb∈H∩a.�en
b⊆acc(H∩a)= a. It followsthat H∩b=(H∩a)∩b. Furthermore,
since H is a historywe have
b= acc(H∩b)=acc((H∩a)∩b) ,
whichshowsthat H∩a is a history.
(c) Letb∈S.�e class
a∶={s∈H∣b∈sorb⊆s}
is nonempty becauseb∈S= acc(H). By�eorem2.7, it has a minimal
elements∈a.
If b∈s=acc(H∩s), there is some set z∈H∩s suchthat b∈z or
b⊆z. It followsthat z∈a. But z∈s∩a impliesthats is not a minimal
element of a. Contradiction.
�erefore, b∉s which implies, by definition of a, that b⊆s. For
transitivity, notethatx∈ b implies
x∈ b⊆s= acc(H∩s)⊆acc(H)= S.
For hereditarity, let x ⊆ b. �en x ⊆ b⊆s∈H, which implies x ∈
acc(H)=S.
(d) By (c) we know that x⊆s∈S implies x∈S. For the other direction,
suppose that x∈S= acc(H). �ere is some set s∈H such that
x∈ s or x⊆ s. By (a), (b), and (c) it followsthat s∈S, s is a stage, and
s is hereditary and transitive. By transitivity, if x∈ s then x⊆s. Consequently,
we havex⊆ s∈Sin both cases andthe claim follows.
(e) By (d),we have S= acc(H(S)). It remains to show that H(S) is
a history. By (c), every element s∈H(S) is hereditary and transitive.
Furthermore,since S istransitive we haves⊆Sand it followsthat
H(S)∩s={x∈ s∣xis astage}.
Sincesis astagewe know by(d)thats= acc(H(S)∩s). ◻
16

2. Stagesand histories
Notethat, by(a) and(b) above,we have H⊆ H(S), for all histories H
of S. In fact, H(S) is the only history of S but we need some further
results beforewe canprovethis.
Exercise 2.4. Prove, by induction on n, that{HF0,... ,HFn−1} is a historywithstage
HFn.
Exercise 2.5. Construct a hereditarytransitivity setathat is not astage.
Hint. It issufficientto consider sets HFn⊂ a⊂ HFn+1, for asmall n.
A�erwe haveseen howto definestageswe nowprovethatthey form
astrictly increasingsequence S0⊆ S1⊆....Togetherwith�eorem2.7
it followsthatthe class of all stages iswell-ordered bythe membership
relation∈(seeSection a3.1).
�eorem 2.10. IfS andT arestagesthataresetsthenwe have
S∈T or S=T or T∈ S .
Proof. Supposethatthere arestages S and T suchthat
(∗) S∉T, S≠T , and T∉ S.
Define
A∶={s∣sis astage andthere issomestage t suchthat
s and t satisfy(∗)}.
By�eorem2.7,the class A has a minimal element S0. Define
B∶={t∣tis astagesuchthat S0 and t satisfy(∗)}.
Againthere is a minimal element T0∈ B.
Ifwe canshowthat H(S0)=H(T0), it followsthat
S0= acc(H(S0))=acc(H(T0))= T0
17

a1. Basicsettheory
in contradiction to our choice of S0 and T0.
Lets∈S0 be astage.�ens≠T0sinceT0∉ S0. Furthermore,we have
T0∉ s since, otherwise, transitivity of S0 would imply that T0∈ S0. By
minimality of S0 it follows that s and T0 do not satisfy(∗). �erefore,
we haves∈T0.
We haveshownthat H(S0)⊆H(T0). Asymmetric argumentshows
that H(T0)⊆H(S0). Hence,we have H(S0)=H(T0) as desired. ◻
Lemma 2.11. Let S and T bestagesthataresets.
(a) S∉ S
(b) S⊆ T ifand only ifS∈ T orS=T.
(c) S⊆ T or T⊆ S.
(d) S⊂ T if,and only if, S∈T.
Proof. (a)Suppose otherwise. LetXbethe class of allstagesssuchthat
s∈s. By �eorem 2.7, X has a minimal element s, that is, an element
suchthats∩ X=∅. Buts∈s∩X. Contradiction.
(b) If S=Tthen S⊆T, and if S∈Tthen S⊆T, bytransitivity of T.
Conversely, if neither S=Tnor S∈ T then �eorem 2.10 impliesthat
T∈ S. IfS⊆T then T∈ S⊆Twould contradict (a).
(c) If S⊈Tthen (b) impliesthat S∉Tand S≠T. By �eorem 2.10,
it followsthatT∈ S which, again by(b), impliesT⊆ S.
(d) We have S⊂Tiff S⊆Tand S≠T. By (a) and (b), the latter is
equivalent toS∈ T. ◻
3. �ecumulative hierarchy
Intheprevioussectionwe haveseenthatwe can arrange allstages in an
increasing sequence
S0⊂ S1⊂⋅⋅⋅⊂Sα⊂⋯ ,
whichwewill call thecumulative hierarchy. If S∈Tarestages then we
willsaythatS isearlierthan T, orthat T is laterthan S.
18

3. �ecumulative hierarchy
From the axioms we have available we cannot prove that there actually
are any stages. We introduce a new axiom which ensures that
enoughstages are available.
Axiom of Creation. Foreveryset athere isaset S∋awhich isastage.
Inparticular, this axiom impliesthat
◆ for every stage S that is aset, there exists a later stage T∋ S that
is also aset.
◆ theuniverseSistheunion of allstages.
Of course, evenwiththis new axiomwe mightstill haveS=∅. But if at
least oneset exists,we can nowprovethat HF⊆S. Inparticular, S=HF
satisfies all axiomswe have introduced so far.
Exercise 3.1. ProvethatSis astagewith history
H(S)={S∣Sis astage}.
Definition 3.1. (a) Wesay that a stage T is the successor of the stage S
if S∈Tand there exists nostage T ′ suchthat S∈T ′ ∈ T. A nonempty
stage is a limit if it is notthesuccessor ofsome otherstage.
(b) Let A be a class. We denote by S(A) the earliest stage such that
A⊆ S(A).
NotethatS(A) iswell-defined by�eorem2.7.We haveS(s)= s, for
every stage s, in particular, S(∅)=∅. �e stages S and HF are limits
and HFn+1 isthesuccessor ofthestage HFn.
Lemma 3.2. a∈bimplies S(a)∈ S(b).
Proof. Since a∈b⊆S(b)=acc(H(S(b))) it followsthatthere issome
stage s∈S(b) such that a∈s or a⊆s. In particular, S(a) is not later
than s which implies that S(a)⊆s∈S(b). As S(b) is hereditary we
therefore have S(a)∈ S(b). ◻
Lemma 3.3. S isthe onlystagethat isaproperclass.
19

a1. Basicsettheory
Proof. Let S be a stage. If S≠ S, there is some set a∈S∖S. Hence,
S(a)∉ S which impliesthat
T∉ H(S) , for allstages T⊇ S(a).
By Lemma2.9(e) and�eorem2.10,we have
H(S)⊆{T∣ T is astagewith T∈ S(a)}= H(S(a)).
Inparticular, H(S) is aset,which impliesthatso isS= acc(H(S)). ◻
Lemma 3.4. Let Abeaclass.�e followingstatements areequivalent:
(1) A isaproperclass.
(2) S(A) isaproperclass.
(3) S(A)=S.
Proof. (3)⇒(1) By the Axiom of Creation, ifA is a set then so isS(A).
(1)⇒(2) IfS(A) is a set then A⊆ S(A) implies that
A={x∈ S(A)∣ x∈ A}
is also a set.
(2)⇒(3) follows by Lemma 3.3. ◻
With the Axiom of Creationwe are finally able to prove most ‘obvious’
properties ofsetssuchthat noset is an element of itself orthattheunion
ofsets is aset.
Lemma 3.5. If a isasetthen a∉a.
Proof. Supposethatthere existssomesetsuchthat a∈a.�en a∈a⊆
S(a) and, by Lemma 2.9(d), there is some stage s∈S(a) with a⊆s.
�is contradicts the minimality of S(a). ◻
�eorem 3.6. Every nonemptyclassA hasaminimalelement.
20

3. �ecumulative hierarchy
Proof. By �eorem 2.7, we can choose some element b∈A such that
S(b) is minimal. We claim that b is a minimal element of A. Suppose
otherwise.�enthere existssome elementx∈ A∩b.Sincex∈ b⊆S(b),
Lemma2.9(d) impliesthatthere issomestages∈S(b)suchthat x⊆ s.
Hence, x is an element of A with S(x)∈ S(b) in contradiction to the
choice ofb. ◻
Wewillsee inSection a3.1 that �eorem 3.6 implies that there are no
infinite descending sequences a0∋a1∋... ofsets. (Ifsuch asequence
existsthentheset{a0,a1,...} has no minimal element.)
Example. By induction on n, ittrivially followsthat, if a0∋⋯∋ ak−1 is
asequence of setsstarting with a0∈ HFn,then k

a1. Basicsettheory
Proof. Let S0 and S1 be stages such that a∈S0 and b∈S1. We know
thatS0⊆S1 orS1⊆S0. By choosing eitherS0 orS1we can find astageS
suchthat S0⊆Sand S1⊆ S. Bytransitivity of S it followsthat
⋃a={x∈ S∣x∈ b forsomeb∈a},
a∪b={x∈ S∣x∈ a or x∈ b},
{a}={x∈ S∣x=a},
and ℘(a)={b∈S∣ b⊆a}. ◻
Corollary 3.9. If a0,... ,an−1 aresetsthenso is
{a0,... ,an−1}={a0}∪⋅⋅⋅∪{an−1}.
Inparticular,every finiteclass isaset.
�e next definitionprovides ausefultoolwhichsometimes allowsus
to replace a proper class A by a set a. Instead of taking every element
x∈Awe only consider thosesuchthat S(x) is minimal.
Definition 3.10. �ecut of a classA istheset
cutA∶={x∈ A∣ S(x)⊆ S(y) for all y∈A}.
Exercise 3.2. What are cutS and cut{x∣a∈x}?
Lemma 3.11. Everyclass ofthe form cutA isaset.
Proof. IfA=∅then cutA=∅. Otherwise, choose an arbitraryseta∈ A.
�en cutA⊆ S(a)which impliesthat cutAis aset. ◻
�e following lemmas clarify the structure of the cumulative hierarchy.
Lemma 3.12. �esuccessor ofastage S is℘(S).
22

3. �ecumulative hierarchy
Proof. By �eorem 2.7, there exists a minimal stage T with S∈T. We
havetoprovethat T=℘(S). a⊆S∈Timplies a∈T since T is hereditary.
Hence,℘(S)⊆ T.
Conversely, if s∈Tis a stage then S∉s because T is the successor
of S. By�eorem2.10, it followsthats∈Sors=S.�is impliess⊆S.
We have shown that s ∈ T iff s ⊆ S, for all stages s. It follows by
Lemma2.9(d)that
T={x∣x⊆ s forsomestages∈T}
={x∣x⊆ s forsomestages⊆S}={x∣ x⊆S}=℘(S). ◻
Lemma 3.13. Let S be a nonempty stage. �e following statements are
equivalent:
(1) S isalimitstage.
(2) S=⋃ H(S).
(3) Foreveryset a∈S,thereexists somestages∈Swith a∈s.
(4) If a∈Sthen℘(a)∈ S.
(5) If a∈Sthen{a}∈ S.
(6) If a⊆Sthen cuta∈S.
Proof. (2)⇒(1) Suppose that S is the successor of a stage T. �en we
have
H(S)={T}∪ H(T).
Sinces⊆T, for all s∈H(T), it followsthat
⋃ H(S)= T≠ S.
(1)⇒(2)Supposethat S is a limitstage. By Lemma2.9(d),we have
S=⋃{℘(s)∣ s∈H(S)}
=⋃{t∣tisthesuccessor ofsomestage s∈H(S)}
=⋃{t∣t∈H(S)}
=⋃ H(S).
23

a1. Basicsettheory
(1)⇒(3) Suppose that S is a limit and let a∈S. By Lemma 2.9(d),
there issomestage s∈Swith a⊆s. Hence, a∈℘(s).Since T∶=℘(s) is
thesuccessor ofswe haveT∈ S.
(3)⇒(4) For each a∈S, there is some stage s∈Swith a∈s. Since
s is transitive it follows that x⊆a implies x∈ s. Hence,℘(a)⊆s. By
transitivity of S,we obtain℘(a)∈ S.
(4)⇒(5) Ifa∈S then{a}⊆℘(a)∈ S. SinceSis hereditary, it follows
that{a}∈ S.
(5)⇒(1) IfSis no limit, there is some stageT∈ SsuchthatS=℘(T).
By assumption,{T}∈S=℘(T). Hence,{T}⊆ T which implies that
T∈ T. A contradiction.
(3)⇒(6) Letb∶= cuta. Ifa=∅thenb=∅andwe are done. Ifthere
is some element x∈a then, by assumption, we can find a stage s∈S
with x∈ s. By definition, b⊆s, and it followsthatb∈S.
(6)⇒(5) Let a∈S and set b∶={x∈S∣a⊆x}. Clearly, b⊆S. By
assumption, we therefore have c∶= cutb∈S. Hence,{a}⊆ c implies
{a}∈ S. ◻
So far,westill might haveS=∅ orS=HF.To excludethese caseswe
introduce a new axiomwhichstatesthat HF∈S.
Axiom of Infinity. �ereexistsasetthat isalimitstage.
We callthetheory consisting ofthe four axioms
◆ Axiom of Extensionality
◆ Axiom ofSeparation
◆ Axiom of Creation
◆ Axiom of Infinity
basic set theory. Every model of this theory consist of a hierarchy of
stages
S0⊂ S1⊂⋯⊂Sω⊂ Sω+1⊂ ...
whereSn= HFn, for finite n.�e differences between twosuch models
can be classified accordingtotwo axes:the length ofthe hierarchy and
thesize of eachstage.
24

3. �ecumulative hierarchy
LetSandS ′ betwo modelswithstages(Sα)α

a1. Basicsettheory
26

a2. Relations
1. Relationsand functions
With basicsettheory availablewe can define most ofthe conceptsused
in mathematics. �esimplest one isthe notion of an ordered pair. �e
characteristic property ofsuchpairs isthat⟨a,b⟩=⟨c,d⟩ implies a=c
and b=d.
Definition1.1. (a) Let a andbbesets.�e ordered pair⟨a,b⟩ istheset
⟨a,b⟩∶={{a},{a,b}}.
(b) Let A and B be classes. �e cartesian product of A and B is the
class
A×B∶={c∣c=⟨a,b⟩ forsome a∈A and b∈B}.
Letusshowthat orderedpairs havethe desiredproperty.
Lemma1.2. If{a,b}={a,c}thenb=c.
Proof. We have b∈{a,b}={a,c}. Hence,b=a or b=c. Inthe latter
case we are done. Otherwise, we have c∈{a,c}={a,b}={b}which
impliesthat c=b. ◻
Lemma1.3. If⟨a,b⟩=⟨c,d⟩then a=candb=d.
Proof. Supposethat⟨a,b⟩=⟨c,d⟩.
{a}∈{{a},{a,b}}={{c},{c,d}}
logic, algebra & geometry2012-08-08 — ©achim blumensath 27

a2. Relations
implies{a}={c} or{a}={c,d}. Inthe latter case,we have a=c=d.
In both cases, wetherefore have{a}={c}. Bythepreceding lemma, it
follows that{a,b}={c,d} and, applying the lemma again, we obtain
b=d. ◻
Remark. �e above definition of an orderedpair⟨a,b⟩ does onlywork
for sets. Nevertheless we will use also pairs⟨A,B⟩ where A or B are
proper classes.�ere areseveral waysto makesuch an expressionwelldefined.
Asimple one isto define
⟨A,B⟩∶=({[0]}×A)∪({[1]}×B) (= A⊍B)
whenever at least one ofA andBis aproper class.(�e operation⊍will
be defined more generally in the next section.) It is easy to check that
withthis definitiontheterm⟨A,B⟩ hastheproperties of an orderedpair,
that is, A⊍B=C⊍DimpliesA= C and B=D.
Definition1.4. (a) Forsets a0,... ,an we define inductively
⟨⟩∶=∅ , ⟨a0⟩∶= a0 ,
and ⟨a0,... ,an⟩∶=⟨⟨a0,... ,an−1⟩,an⟩.
We call⟨a0,... ,an−1⟩ atuple of length n.⟨⟩ istheemptytuple.
(b) For a class A,we define its n-thpower by
A 0 ∶={⟨⟩} , A 1 ∶= A , and A n+1 ∶= A n ×A , for n> 1.
Definition1.5. Arelation, or apredicate, ofarity n is asubclassR⊆S n .
IfR⊆A n , forsome class A,wesaythatRis over A.
Notethat∅and{⟨⟩} arethe onlyrelations of arity0. In logicthey are
usually interpreted as false andtrue. Arelation of arity 1 over A is just a
subclassR⊆A.
Definition1.6. LetRbe a binary relation.�edomain ofRisthe class
28
domR∶={a∣⟨a,b⟩∈ R forsomeb},

and itsrange is
rngR∶={b∣⟨a,b⟩∈ R forsome a}.
�e field of R is domR∪rngR.
1. Relationsand functions
Inparticular, domR andrngR arethesmallest classessuchthat
R⊆domR×rngR.
Definition 1.7. (a) A binary relation R is called functional if, for every
a∈ domR,there exists exactly onesetbsuchthat⟨a,b⟩∈ R.We denote
thisunique elementbby R(a). Hence,we canwrite R as
R={⟨a,R(a)⟩∣ a∈ domR}.
A functional relation R⊆A×Bis also called apartial function from A
toB.
(b) A function from A to B is a functional relation f ⊆ A× B with
dom f= A and rng f⊆ B. Functions are also called maps or mappings.
Wewrite f∶ A→ Bto denotethe factthat f is a function fromAtoB.
A function ofarity n is a function ofthe form
f∶ A0×⋯×An−1→ B.
We will write x↦y to denote the function f such that f(x)= y.
(Usually, y will be an expression depending onx.)
(c) For asetaand a classB,we denote byB a the class of all functions
f∶ a→B.
Remark. A 0-ary function f ∶ A 0 → B is uniquely determined by the
value f(⟨⟩).Wewill callsuch functionsconstants and identifythemwith
their only value.
Sometimes wewrite binary relations and functions in infix notation,
that is, for arelationR∈A×A,wewritea R b instead of⟨a,b⟩∈ R and,
for f∶ A×A→ A,wewrite a f b instead of f(a,b).
29

a2. Relations
Definition 1.8. (a) For every class A, we define the identity function
idA∶A→ A by idA(a)∶= a.
(b) If R⊆A× B and S⊆B×C are relations, we can define their
composition S○R∶A×C by
S○R∶={⟨a,c⟩∣there issomeb∈Bsuchthat
⟨a,b⟩∈ R and⟨b,c⟩∈ S}.
(Note the reversal of the ordering.) In particular, if f ∶ A→ B and g∶
B→C are functionsthen
(g○ f)(x)∶= g( f(x)).
(c)�e inverse of arelation R⊆A×B istherelation
R −1 ∶={⟨b,a⟩∣⟨a,b⟩∈ R}.
Inparticular, a function g∶B→A isthe inverse ofthe function f∶ A→
B if
g( f(a))= a and f(g(b))=b , for all a∈A and b∈B ,
that is, if g○ f= idA and f○ g= idB.
Forb∈B,wewillwrite
R −1 (b)∶={a∣⟨a,b⟩∈ R}.
Notethat, ifR −1 is a function,we have already defined
R −1 (b)∶= a fortheunique asuchthat⟨a,b⟩∈ R.
Itshould always be clear fromthe contextwhich ofthesetwo definitions
we have in mindwhenwewriteR −1 (b).
(d)�erestriction of arelation R⊆A×Bto a classC istherelation
30
R∣C∶= R∩(C×C).

Its le�restriction is
R↾C∶= R∩(C×B) .
1. Relationsand functions
(e) �e image of a class C under a binary relation R⊆A× B is the
class
R[C]∶=rng(R↾C).
Remark. �esetA A togetherwiththe operation○ forms a monoid,that
is,○isassociative
f○(g○h)=( f○ g)○ h , for all f , g, h∈A A ,
andthere exists a neutralelement
idA○ f=f and f○ idA= f for all f∈ A A .
Exercise1.1. Is ittruethatR −1 ○R=idA, for allrelations R⊆A×B?
Exercise1.2. Provethat○is associative andthat idA is a neutral element.
Definition1.9. Let f∶ A→ B be a function.
(a) f is injective ifthere is nopair a,a ′ ∈ A of distinct elementssuch
that f(a)= f(a ′ ).
(b) f issurjective ifrng f= B.
(c) f is called bijective if it is both injective andsurjective.
Lemma1.10. Let f∶ A→ Bbeafunction.
(a) �e followingstatements areequivalent:
(1) f isbijective.
(2) f −1 isafunction B→A.
(3) �ereexists a function g∶B→Asuchthat g○ f= idA and
f○ g= idB.
31

a2. Relations
(b) �e followingstatements areequivalent:
(1) f is injective.
(2) f○ g= f○ h implies g= h, forall functions g, h∶C→ A.
(3) A=∅ or there exists some function g∶B→A such that
g○ f= idA.
(4) f −1 [ f[X]]= X, forall X⊆A.
(c) �e followingstatements areequivalent:
(1) f issurjective.
(2) g○ f= h○ f implies g= h, forall functions g, h∶B→C.
(3) f[ f −1 [Y]]=Y, forallY⊆ B.
(d) Ifthere exists some function g∶B→Asuchthat f○ g= idB then
f issurjective.
Proof. (a) (1)⇒(2) Let b∈B. Since f is surjective there exists some
a∈Asuchthat f(a)=b. Ifa ′ ∈ A issome elementwith f(a ′ )=bthen
the injectivity of f implies that a ′ = a. We have shown that, for every
element b∈B, there is a unique a∈A such that f −1 (b)= a. Hence,
f −1 is functional and dom f −1 = B.
(2)⇒(3) f −1 ∶ B→A is a function and we have f −1 ○ f = idA and
f○ f −1 = idB.
(3)⇒(1) If f(a)= f(b), for a,b∈ A, then
a= idA(a)=(g○ f)(a)=(g○ f)(b)=idA(b)=b .
Consequently, f is injective. To show that it is also surjective let b∈B.
Setting a∶= g(b)we have
f(a)=( f○ g)(b)=idB(b)=b.
Hence,b∈rng f.
(b)(1)⇒(4) Let X⊆ A. For every a∈X,we have f(a)∈ f[X] and,
therefore, a∈ f −1 [ f[X]]. Consequently, X⊆ f −1 [ f[X]]. Conversely,
32

1. Relationsand functions
supposethat a∈ f −1 [ f[X]] and set b∶= f(a).Since b∈ f[X] there is
some c∈Xwith f(c)=b. As f is injective this impliesthat a=c∈X.
(4)⇒(3) If A=∅ then there is nothing to do. Hence, assume that
A≠∅.We define g as follows. For everyb∈rng f ,there issome element
a∈Awith f(a)=b.Since f −1 (b)= f −1 [ f[{a}]]={a} it followsthat
this element a isunique. Hence, fixing a0∈Awe can define g by
⎧⎪ a if f
g(b)∶= ⎨
⎪⎩
−1 (b)={a} ,
a0 ifb∉rng f .
(3)⇒(2) IfA=∅, there are no functionsC→A andthe claim holds
trivially. Hence, assume that A≠∅ and let k be a function such that
k○ f= idA.�en f○ g= f○ h implies
g= idA○g=k○ f○ g=k○ f○ h= idA○h=h.
(2)⇒(1)Supposethat f is not injective. �enthere aretwo elements
a,b∈A with a≠b such that f(a)= f(b). Let C∶=[1]={0} be a
set with a single element and define g, h∶C→A by g(0)∶= a and
h(0)∶= b.�en g≠h but f○ g= f○ h.
(c) (1)⇒(2) Suppose that g≠h. �ere is some element b∈B with
g(b)≠ h(b). Since f is surjective we can find an element a∈A with
f(a)=b. Hence,(g○ f)(a)= g(b)≠ h(b)=(h○ f)(a).
(2)⇒(1)Supposethat f is notsurjective.�enthere issome element
b∈B∖rng f. LetC∶=[2]={0, 1} be asetwithtwo elements and define
g, h∶B→Cby
⎧⎪ 1 if x= b ,
g(x)∶= ⎨
⎪⎩
0 otherwise,
and h(x)∶=0 , for all x∈ B.
�enwe have g≠h but g○ f= h○ f.
(3)⇒(1) f[ f −1 [B]]=B implies that rng f= B.
(1)⇒(3) Let Y⊆ B. If b∈ f[ f −1 [Y]] then there is some a∈ f −1 [Y]
with f(a)=b. Hence, a∈ f −1 [Y] implies that b= f(a)∈Y. Consequently,
we have f[ f −1 [Y]]⊆Y.
33

a2. Relations
For the converse, let b∈Y. Since f is surjective there is some a∈
A with f(a)=b. Hence, a∈ f −1 [Y] and it follows that b= f(a)∈
f[ f −1 [Y]].
(d) Let k be a functionsuchthat f○k= idB.�en g○ f= h○ f implies
g=g○ idB=g○ f○ k=h○ f○ k=h○ idB= h.
By(c), it followsthat f issurjective. ◻
Remark. �e converse of(d) also holds butwe cannotprove itwithout
the Axiom of Choice, which we will introduce in Section a4.1 below.
Actually one can prove that the Axiom of Choice is equivalent to the
claimthat, for everysurjective function f ,there existssome function g
with f○ g= id.
Remark. �esubset of all bijective functions f∶ A→ A forms a group
since, bythepreceding lemma, every element f has an inverse f −1 .
Exercise1.3. Let f∶ A→ B and g∶B→Cbe functions. Prove that, if
f and g are(a) injective, (b)surjective, or(c) bijective thenso is g○ f.
We concludethis section with two important results about the existence
of functions. �e first one can be used to prove that there exists
a bijection between two given sets without constructing this function
explicitly.
Lemma 1.11. Let A⊆ B⊆C be sets. If there exists a bijective function
f∶ C→A,there isalsoabijection g∶ C→B.
Proof. Let
Z∶=⋂{X⊆ C∣ C∖B⊆ X and f[X]⊆X}.
�enC∖B⊆Zand f[Z]⊆ Z.We claimthat
34
⎧⎪ f(x) if x∈Z ,
g(x)∶= ⎨
⎪⎩
x otherwise,

C
B
A
Z
Y
1. Relationsand functions
f
id
Figure 1..�eproof of Lemma 1.11.
Z∩B
is the desired bijection g∶ C→B.
LetY∶= C∖Z be the complement ofZ. Since g[Y]⊆ Y and g[Z]⊆ Z
it issufficienttoshowthattherestrictions g↾Y∶Y→ Y and g↾Z∶Z→
Z∩B are bijections. Clearly, g↾Y= idY is bijective and g↾Z= f↾Z is
injective. �erefore,we only needtoprovethat f[Z]= Z∩B.
By definition of Z,we have f[Z]⊆Z∩rng f⊆ Z∩B. For the other
inclusion, supposethat there exists some element a∈(Z∩B)∖ f[Z].
Since a∈B theset X∶= Z∖{a}satisfies C∖B⊆Xand f[X]⊆ X. By
definition of Z, it followsthat Z⊆X. Contradiction. ◻
�eorem1.12(Bernstein). Ifthereare injective functions f∶ A→ Band
g∶B→Athenthere existsabijective function h∶A→ B.
Proof. We have g[ f[A]]⊆g[B]⊆A. Since f and g are injective so
is their composition g○ f. When regarded as function g○ f ∶ A→
g[ f[A]] it is alsosurjective. Hence, bythepreceding lemma,there exists
a bijective mapping h∶A→ g[B].Since k∶= g −1 ↾ g[B]∶ g[B]→ B is
bijective it followsthatso is k○h∶A→ B. ◻
�e second result deals with functions between a set and its power
set.
�eorem1.13(Cantor). Foreveryset a,thereexistsan injective function
a→℘(a)but nosurjective one.
Y
35

a2. Relations
Proof. �e function f∶ a→℘(a)with f(x)∶={x} is injective.
For a contradiction, suppose that there is also a surjective function
f∶ a→℘(a).We definetheset
z∶={x∈ a∣x∉ f(x)}⊆ a.
Since f is surjective there is some element b∈a with f(b)=z. By
definition ofz,we have
b∈ z iff b∉ f(b)=z.
A contradiction. ◻
Corollary1.14. Forallsets a,thereare no injective functions℘(a)→ a.
Proof. Suppose that f ∶℘(a)→ a is injective. We define a function
g∶a→℘(a) by
⎧⎪ f
g(x)∶= ⎨
⎪⎩
−1 (x) if x∈rng f ,
∅ otherwise.
Note that g is well-defined since f is injective. Furthermore, we have
g○ f= id ℘(A). Hence, Lemma 1.10(d) impliesthat g issurjective. �is
contradicts the�eorem of Cantor. ◻
2. Productsandunions
So far, we have defined cartesian products of finitely many sets and
tuples of finite length. In this section we will show how to generalise
these definitions to infinitely many factors.
Remark. (a) �ere is a canonical bijection π∶A [n] → A n between the
setA [n] of all functions[n]→A andthe n-thpowerA n of A.πmaps a
function f∶[n]→Atothetuple
36
π( f)∶=⟨ f(0),..., f(n− 1)⟩ ,

2. Productsandunions
and its inverseπ −1 maps a tuple⟨a0,... ,an−1⟩tothe function f∶[n]→
Awith f(i)= ai.
(b)�ere is also a canonical bijection π∶(A×B)×C→A×(B×C)
defined by
π⟨⟨a,b⟩,c⟩∶=⟨a,⟨b,c⟩⟩.
(c) Finally, letus define a canonical bijection π∶A B×C →(A C ) B that
maps a function f∶ B×C→Atothe function g∶B→A C with
g(b)∶= hb where hb(c)∶= f(b,c) , forb∈B, c∈C.
Inthetheory ofprogramming languages thistransformation of a function
B×C→A into a function B→A C is calledcurrying.
Part (a) of the above remark gives a hint on how to generalise finite
tuples. Atuple of length n correspondsto a map[n]→A.�erefore,we
define an infinitetuple as mapN→ A.
Definition 2.1. (a) Let A be a class and I aset. A function f∶ I→A is
called asequence, or family, over I. If f(i)= ai then we alsowrite f in
the form(ai)i∈I.
(b) Let I be aset,(Ai)i∈I asequence ofsets, and A∶=⋃{Ai∣ i∈I}
theirunion.�eproduct of(Ai)i∈I isthe class
∏Ai∶={ f∈ A
i∈I
I ∣ f(i)∈ Ai for all i}.
(c) Let(Ai)i∈I be asequence of sets and k∈I.�e projection tothe
k-th coordinate isthe map
pr k ∶∏ i∈I
Ai→ Ak with pr k ( f)∶= f(k).
Remark. (a) IfAi= A, for all i∈I,then∏i∈I Ai= A I .
(b) As we have seen above there is a canonical bijection between
A0×A1 and∏i∈[2]Ai. Inthe followingwewill not distinguish between
thesesets.
37

a2. Relations
Letus introducesome notation and conventionsregardingsequences.
To indicate that a certain variable refers to a sequence we will write it
with a bar ā. Ifthesequence is over I,the components of ā will always
be(ai)i∈I. Sometimes we will not distinguish between a sequence ā=
(ai)i∈I and its range rngā={ai∣ i∈I}. Inparticular, wewrite ā∪ ¯ b
instead ofrngā∪rng ¯ b and, ifwe do notwanttospecifythe indexset I,
wewill write ā⊆A instead of ā∈A I . Finally, for a function f∶ A→ B,
wewrite f(ā)to denotethesequence( f(ai))i∈I.
Lemma 2.2. Let A be a set and(Bi)i∈I a sequence of sets. For every sequence(
fi)i∈I of functions fi ∶ A→ Bi there exists a unique function
g∶ A→∏i Bi suchthat
pr i ○ g= fi , forall i∈I.
Proof. �e function
g(a)∶=( fi(a))i∈I
has obviously the desiredproperties.We havetoshowthat it isunique.
Let h∶A→∏i Bi be another such function. If g≠h, there is some
element a∈A such that g(a)≠ h(a). Let(bi)i∈I∶= h(a). For every
i∈I,we have
bi=(pr i ○ h)(a)= fi(a).
Hence g(a)=( fi(a))i=(bi)i= h(a). A contradiction. ◻
Definition 2.3. �e disjoint union of a sequence(Ai)i∈I of sets is the
class
⊍Ai∶={⟨i,a⟩∣ i∈I, a∈Ai}.
i∈I
Similarly, if A and B are classesthenwe can definetheir disjoint union
as
38
A⊍B∶=({[0]}×A)∪({[1]}×B).

�e k-th insertion is the canonical map
ink∶ Ak→⊍ Ai with ink(a)∶=⟨k,a⟩ .
i∈I
Remark. IfAi= A, for all i∈I, then⊍i∈I Ai= I×A.
3. Graphsandpartial orders
Lemma 2.4. Let B be a set and(Ai)i∈I a sequence of sets. For every sequence(
fi)i∈I of functions fi ∶ Ai → B there exists a unique function
g∶⊍i Ai→ Bsuchthat
g○ ini=fi , forall i∈I .
Proof. �e function
g⟨i,a⟩∶= fi(a)
has obviously the desiredproperties.We havetoshowthat it isunique.
Let h∶⊍i Ai→ B be anothersuch function. If g≠hthenthere issome
element⟨k,a⟩∈⊍i Ai suchthat g⟨k,a⟩≠ h⟨k,a⟩.We have
h⟨k,a⟩=(h○ ink)(a)= fk(a)= g⟨k,a⟩.
A contradiction. ◻
3. Graphsandpartial orders
When consideringrelations it is frequently necessarytospecifythesets
they are over.
Definition 3.1. A graph is a pair⟨A,R⟩ where R⊆A× A is a binary
relation onA.
More generally one can considersetstogetherwithseveralrelations and
functions. �iswill leadtothe notion of astructure in Chapter b1.
Definition 3.2. Let⟨A,R⟩ be a graph.
39

a2. Relations
(a) R isreflexive if⟨a,a⟩∈ R, for all a∈A.
(b) R is irreflexive if⟨a,a⟩∉ R, for all a∈A.
(c) R issymmetric ifwe have⟨a,b⟩∈ R if, and only if,⟨b,a⟩∈ R, for
all a,b∈A.
(d) R isantisymmetric if⟨a,b⟩∈ R and⟨b,a⟩∈ R implies a=b.
(e) R istransitive if⟨a,b⟩∈ R and⟨b,c⟩∈ R implies⟨a,c⟩∈ R, for all
a,b,c∈A.
Note that, for the definition of reflexivity, it is important to specify
the setA. If⟨A,R⟩ is reflexive and A⊂ B then⟨B,R⟩ is not reflexive.
Example. (a) �e relation A×A is reflexive, symmetric, and transitive.
It is irreflexive if, and only if,A=∅, and it is antisymmetric if, and only
if,A contains at most one element.
(b) �e diagonal idA={⟨a,a⟩∣ a∈A} is reflexive, symmetric, antisymmetric,
and transitive. It is irreflexive if, and only if,A=∅.
(c) �e empty relation∅⊆A×Ais irreflexive, symmetric, antisymmetric,
and transitive. It is reflexive if, and only if, A=∅.
Definition 3.3. (a) A (non-strict) partial order is a graph⟨A,≤⟩where
≤ is reflexive, transitive, and antisymmetric.
(b) A strict partial order is a graph⟨A,

3. Graphsandpartial orders
Similarly, if

a2. Relations
the greatest element of X by maxA X and the least element by minAX,
providedthese elements exist.
(d) Let X⊆A. Wesay that a is an upper bound of X if x≤a, for all
x∈X. If a is an upper bound of X and a≤b, for every other upper
boundbof X,then a isthe leastupperbound, orsupremum, of X. Ifthe
leastupper bound of X exists,we denote it bysup A X.
�e notion of a (greatest) lower bound is defined analogously. �e
greatest lower bound is also called the infimum of X. We denote it by
infA X. Ifthe order A isunderstoodwewill omitthesubscript A andwe
justwritesupX and inf X.
(e) A linearly orderedsubsetC⊆A is called achain.
Example. (a) Let Q∶=⟨Q,≤⟩. �e set I∶={x∈ Q∣x< √ 2} is an
initial segment of Q. Every rational number y> √ 2 is anupper bound
of I but I has no leastupper bound.
(b) Consider⟨N,∣⟩. Its least element isthe number 1 and its greatest
element is 0. �e least upper bound of two elements[k],[m]∈N is
their least common multiple lcm(k, m), andtheir greatest lower bound
istheir greatest common divisor gcd(k, m).�eset P⊆N of all prime
numbers has the least upper bound 0 and the greatest lower bound 1.
�eset{2 n ∣ n∈N} of allpowers oftwo forms a chain.
Exercise 3.1. Consider⟨B,⊆⟩where
B∶={X⊆N∣ X is finite orN∖X is finite}.
(a) Construct aset X⊆Bthat has no minimal element.
(b) Construct aset X⊆Bwith lower bounds butwithout infimum.
Lemma 3.5. Let⟨A,≤⟩beapartial order. IfA isaset,the followingstatementsareequivalent:
42
(1) Everysubset X⊆A hasasupremum.
(2) Everysubset X⊆A hasan infimum.

3. Graphsandpartial orders
Proof. We only prove (1)⇒(2). �e other direction follows in exactly
thesameway. Let X⊆A andset
C∶={a∈A∣ a is a lower bound of X}.
By assumption, c∶= supC exists. We claim that inf X=c. Let b∈X.
By definition, we have a≤b, for all a∈C. Hence, b is anupper bound
of C andwe haveb≥supC= c. Asb was arbitrary it followsthat c is a
lower bound of X. If a is an arbitrary lower bound of X,we have a∈C,
which implies that a≤c. Consequently, c is the greatest lower bound
of X. ◻
Definition 3.6. Apartial order⟨A,≤⟩ iscomplete if everysubset X⊆A
has an infimum and asupremum.
Remark. Every complete partial order has a least element�∶= sup∅
and a greatest element⊺∶= inf∅.
Example. (a) Let A be aset. �epartial order⟨℘(A),⊆⟩ is complete. If
X⊆℘(A)then
supX=⋃X∈℘(A) and inf X=⋂X∈℘(A).
(b)�e order⟨R,≤⟩ is complete.⟨Q,≤⟩ is notsincetheset
{x∈ Q∣x≤π}
has no leastupper bound inQ.
(c) �e order⟨N,≤⟩ is not complete since inf∅ and supN do not
exist.
(d) Let A=⟨A,≤⟩ be an arbitrary partial order. We can construct a
completepartial order C=⟨C,⊆⟩ containing A as follows. LetC⊆℘(A)
betheset of all initial segments of A ordered by inclusion. �e desired
embedding f∶ A→ C is given by f(a)∶=⇓Aa.
43

a2. Relations
Nextwe turntothestudy of functions betweenpartial orders. Inparticular,wewill
consider functions f∶ A→ A mapping onepartial order
into itself.Tosimplify notation, wewillwrite
f∶ A→B ,
for partial orders A=⟨A,≤A⟩ and B=⟨B,≤B⟩, to denote that f is a
function f∶ A→ B.
Definition 3.7. Let A=⟨A,≤A⟩ and B=⟨B,≤B⟩ bepartial orders.
(a) A function f∶ A→ B is increasing if
a≤A b implies f(a)≤B f(b) , for all a,b∈ A ,
and f isstrictly increasing if
a

3. Graphsandpartial orders
Lemma 3.8. Let⟨A,≤A⟩and⟨B,≤B⟩bepartial ordersand h∶A→ Ban
increasing function. Let C⊆Aand a∈A.
(a) If a isanupperbound ofCthen h(a) isanupperbound of h[C].
(b) If a isalowerbound ofC then h(a) isalowerbound of h[C].
Lemma 3.9. Let⟨A,≤A⟩and⟨B,≤B⟩bepartial ordersand h∶A→ Ban
embedding. LetC⊆Aand a∈A.
(a) h(a)=sup h[C] implies a=supC.
(b) h(a)= inf h[C] implies a= inf C.
Proof. (a)Since h is an embedding it followsthat h(c)≤B h(a) implies
c≤A a, for c∈C. Hence, a is an upper bound of C. To show that it is
the least one, suppose that b is another upper bound of C. �en c≤A
b, for c∈C, implies h(c)≤B h(b). Hence, h(b) is an upper bound
of h[C].Since h(a) isthe leastsuch bound it followsthat h(a)≤B h(b).
Consequently, we have a≤A b, as desired.
(b) h is also an embedding of⟨A,≥A⟩ into⟨B,≥B⟩. Hence,(b) follows
from(a) byreversing the orders. ◻
Corollary 3.10. Let⟨F,⊆⟩beapartial orderwith F⊆℘(A)andC⊆F.
(a)⋃C∈FimpliessupC=⋃C.
(b)⋂C∈Fimplies inf C=⋂C.
Proof. We can apply Lemma 3.9 to the inclusion map F→℘(A). ◻
Corollary 3.11. Let A=⟨A,≤⟩beapartial order. IfB⊆A isanonempty
setsuchthat
inf AX∈ B and sup A X∈ B , forevery nonempty X⊆B ,
then B∶=⟨B,≤⟩ is a complete partial order where, for every nonempty
subset X⊆B,we have
inf BX= inf AX and sup B X=sup A X.
45

a2. Relations
Proof. If X⊆B is nonemptythen, applying Lemma 3.9 to the inclusion
map B→A, it followsthat
inf BX= infAX and sup B X=sup A X.
Inparticular, inf BXandsup B X exist. Forthe emptyset, it followssimilarlythat
inf B∅=sup B B=sup A B∈B ,
and sup B ∅=inf BB=inf AB∈B.
Consequently, B is complete. ◻
We haveseen that although increasing functions preserve the ordering
of elementsthey do not necessarily preservesupremums and infimums.
Letustake a look at functionsthat do.
Definition 3.12. Let⟨A,≤A⟩ and⟨B,≤B⟩ be partial orders. A function
f ∶ A→ B is continuous if, whenever a nonempty chain C⊆A has a
supremumthen f[C] also has asupremum andwe have
sup f[C]= f(supC).
f is called strictlycontinuous if it is continuous and injective.
Remark. Every (strictly) continuous function is(strictly) increasing.
Exercise 3.4. Provethat continuous functions are increasing.
Example. (a) Let⟨A,≤⟩ bethe linear orderwhereA=N⊍Nand
46
⟨i,a⟩≤⟨k,b⟩ : iff i

4. Fixed pointsandclosure operators
�e function f∶ A→ A∶⟨i,a⟩↦⟨i,a+1⟩ is not continuous. Consider
the initial segment X∶={0}×N=↓⟨1,0⟩⊆ A.We have supX=⟨1,0⟩
but
sup f[X]=⟨1,0⟩

a2. Relations
Figure2.. Fixedpoints of f(x)= 1 4x 3 − 3 4x 2 + 3 4x+3 4
Definition 4.1. Let f ∶ A→A be a function. An element a∈A with
f(a)= a is called a fixed point of f. �e class of all fixed points of f is
denoted by
fix f∶={a∈A∣ f(a)= a}.
We denotethe least and greatest fixedpoint of f , if it exists, by
lfp f∶= min fix f and gfp f∶= max fix f .
Example. (a) Let⟨R,

4. Fixed pointsandclosure operators
hasthe fixedpoints{0},{1},{0, 1}. It has no least fixedpoint.
(d) Consider⟨F,⊆⟩where
F∶={X⊆N∣ X orN∖X is finite}.
�e function f∶ F→ F defined by
⎧⎪ X∪{1+maxX} if X is finite,
f(X)∶= ⎨
⎪⎩
X otherwise,
has fixed points
fix f={X⊆N∣N∖ X is finite} ,
but no least one.
Exercise 4.1. Let A=⟨℘(N),⊆⟩. Construct a function f∶ A→A that
has a least fixedpoint but no greatest one.
Not every function has fixed points. �e next theorem presents an
important special casewherewe always have a least fixed point. InSection
a3.3wewill collect furtherresults aboutthe existence of fixedpoints
and methodsto computethem.
�eorem 4.2 (Knaster, Tarski). Let⟨A,≤⟩ be a complete partial order
where A is a set. Every increasing function f ∶ A→ A has a least fixed
pointandwe have
lfp f= inf{a∈A∣ f(a)≤ a}.
Proof. Set B∶={a∈A∣ f(a)≤ a} and b∶= inf B. For every a∈B,
b≤a implies f(b)≤ f(a)≤ a,since f is increasing. Hence, f(b) is a
lower bound ofBand it followsthat f(b)≤inf B=b.�is impliesthat
f( f(b))≤ f(b) and, by definition ofB, it followsthat f(b)∈ B. Hence,
f(b)≥inf B=b. Consequently, we have f(b)=b andbis a fixedpoint
of f.
Let a be another fixed point of f. �en f(a)= a implies a∈B and
we haveb= inf B≤a. Hence,bisthe least fixedpoint of f. ◻
49

a2. Relations
�eorem4.3. Let⟨A,≤⟩beacompletepartial orderwhereA isasetand
let f∶ A→ Abe increasing.�eset F∶= fix f is nonemptyand F∶=⟨F,≤⟩
formsacompletepartial orderwhere, for X⊆ F,
infF X=sup A {a∈A∣ a≤infAX and f(a)≥ a} ,
sup F X= inf A{a∈A∣ a≥sup A X and f(a)≤ a}.
Proof. We have already shown inthepreceding theorem that F≠∅. It
remains toprove that F is complete. For X⊆ A, let U∶=⇑sup A X⊆A
betheset of allupper bounds of X. If Z⊆U then
sup A Z≥sup A X and inf AZ≥sup A X.
It followsthatthepartial order⟨U,≤⟩ is complete. Furthermore, ifa∈ U
andx∈Xthena≥ximplies f(a)≥ f(x). Hence, f↾U is an increasing
functionU→U. By�eorem 4.2, it followsthat
sup F X= lfp( f↾U)=infA{a∈U∣ f(a)≤ a} ,
as desired. �e claim for infF X follows by applying the equation for
sup F X tothe dual order A op . ◻
Example. Consider a closed interval[a,b]⊆Rofthereal line.
(a)Sincethe order⟨[a,b],

4. Fixed pointsandclosure operators
As a special case of �eorem 4.3we consider complete partial orders
obtained via closure operators.
Definition 4.4. LetA be a class.
(a) Aclosure operator on A is a function c∶℘(A)→℘(A)suchthat,
for all x,y∈℘(A),
◆ x⊆c(x) ,
◆ c(c(x))= c(x) , and
◆ x⊆y impliesc(x)⊆ c(y).
(b) Aset x⊆A isc-closed ifc(x)= x.
(c) A closure operator c has finite character if, for all sets x⊆A, we
have
c(x)=⋃{c(x0)∣ x0⊆ x is finite}.
If c has finite characterwe alsosaythat c isalgebraic.
(d) A closure operatorc istopological ifwe have
◆ c(∅)=∅and
◆ c(x∪ y)=c(x)∪c(y) , for all x,y∈℘(A).
Remark. Letcbe a closure operator on A.
(a) �e class of c-closedsets is fixc=rngc.
(b) Ifthe class A is asetthen it is c-closed.
Example. (a) Let V be a vector space. For X⊆V, let⟪X⟫ be the subspace
ofV spanned by X.�e function X↦⟪X⟫ is a closure operator
with finite character.
(b) Let X be a topological space. For A⊆ X, let c(A) be the topological
closure ofA in X.�encis atopological closure operator.
(c) LetA be aset and a∈A.�e functions c,d∶℘(A)→℘(A)with
c(X)∶= X and d(X)∶= X∪{a}
are closure operators on A.
51

a2. Relations
Exercise 4.2. Let A=⟨A,≤⟩ be apartial order. For X⊆A,we define
c(X)∶={supC∣ C⊆Xis a nonempty chainwithsupremum}.
(a)Provethatthe function c is atopological closure operator on A.
(b) Let B be asecond partial order and d the corresponding closure
operator.Prove that a function f∶ A→B is continuous if, and only if,
every d-closedset X∈ fixd has ac-closedpreimage f −1 [X]∈fixc.
Exercise 4.3. Let⟨A,≤⟩ be apartial order. Forsets X⊆A,we define
U(X)∶={a∈A∣ a is anupper bound of X} ,
L(X)∶={a∈A∣ a is a lower bound of X}.
Provethatthe function c∶X↦L(U(X)) is a closure operator on A.
Lemma 4.5. Let c beaclosure operator on Aandx,y⊆Asets.
(a) c(x)∪c(y)⊆ c(x∪ y).
(b) c(x∪ y)= c(c(x)∪c(y)).
Proof. (a) By monotonicity of c, we have c(x)⊆ c(x∪y) and c(y)⊆
c(x∪ y).
(b) It follows fromx∪y⊆c(x)∪c(y) and(a)that
c(x∪ y)⊆ c(c(x)∪c(y))⊆ c(c(x∪ y))=c(x∪ y). ◻
Lemma 4.6. Let c be a closure operator on A with finite character. For
everychainC⊆ fixc,we have
c(⋃C)=⋃C.
Proof. By definition, we have⋃C⊆c(⋃C). Forthe converse, let x0⊆
⋃C be finite.SinceC is linearly ordered by⊆there existssome element
x∈Cwith x0⊆x. Hence,we have c(x0)⊆c(x)= x⊆⋃C. It follows
that
52
c(⋃C)=⋃{c(x0)∣x0⊆⋃C finite}⊆⋃C. ◻

4. Fixed pointsandclosure operators
If c is a closure operator, the setC ∶= fixc of c-closed sets has the
followingproperties.
Definition 4.7. AsetC⊆℘(A) is called asystem ofclosedsets ifwe have
◆ A∈C and
◆ ⋂Z∈C, for every Z⊆C.
Apair⟨A,C⟩whereC⊆℘(A) is asystem of closedsets is called aclosure
space.
Lemma 4.8. (a) If c isaclosure operator on Athen fixc forms asystem
ofclosedsets.
(b) IfC⊆℘(A) isasystem ofclosedsetsthenthe mapping
c∶X↦⋂{C∈C∣X⊆ C}
definesaclosure operator on Awith fixc=C.
�e followingtheorem statesthatthe family of c-closedsets forms a
completepartial order.We canusethisresulttoprovethat a given partial
order A is complete by defining a closure operatorwhose closedsets
are exactlythe elements of A. An example ofsuch aproof isprovided in
Corollary 4.17.
�eorem 4.9. Let A be a set and c a closure operator on A. �e graph
⟨F,⊆⟩with F∶= fixc forms acomplete partial orderwith
inf X=⋂X and supX=c(⋃X) , forall X⊆ F.
Proof. Since closure operators are increasingwe can apply�eorem 4.3.
By Lemma 4.8 (b), it follows that
supX=⋂{Z⊆A∣ Z⊇⋃X and c(Z)⊆ Z}
=⋂{Z⊆A∣ Z⊇⋃X and c(Z)= Z}
= c(⋃X) ,
53

a2. Relations
and inf X=⋃{Z⊆ A∣ Z⊆⋂X and c(Z)⊇ Z}
=⋃{Z⊆ A∣ Z⊆⋂X}
=⋂X . ◻
Corollary 4.10. Let c be a closure operator on Aand set F∶= fixc.�e
operator c iscontinuous ifweconsider itasafunction
c∶⟨℘(A),⊆⟩→⟨F,⊆⟩ .
Proof. For a nonempty chain X⊆℘(A),we have
c(supX)= c(⋃X)⊆ c(⋃c[X])=supc[X]
⊆sup{c(supX)}= c(supX). ◻
As an application of closure operators we consider equivalence relations.
Definition 4.11. (a) A binaryrelation∼⊆A×A is anequivalencerelation
on A if it isreflexive,symmetric, andtransitive.
(b) Let∼⊆A× A be an equivalence relation. If A is a set, we define
the∼-class of an element a∈A by
[a]∼∶={b∈A∣ b∼a}.
Forproper classes A,weset
[a]∼∶= cut{b∈A∣ b∼a}.
Notethat, despitethe name, a∼-class is always aset.We denotethe class
of all∼-classes by
A/∼∶={[a]∼∣a∈A}.
Example. (a)�e diagonal idA isthesmallest equivalencerelation onA.
�e largest one isthe fullrelation A×A.
(b)�e isomorphismrelation≅ is an equivalencerelation onthe class
of allpartial orders.
54

4. Fixed pointsandclosure operators
Lemma4.12. Let∼beanequivalencerelation on Aand a,b∈ A.�en
a∼b iff [a]∼=[b]∼ iff [a]∼∩[b]∼≠∅.
Remark. LetA be aset. Apartition of A is asetP⊆℘(A) of nonempty
subsets ofAsuchthatA=⋃P and p∩q=∅, for all p,q∈Pwith p≠q.
If∼is an equivalence relation on Athen A/∼ forms apartition on A.
Conversely, given apartition P of A,we can define an equivalence relation∼P
onAwith A/∼P= P bysetting
a∼P b : iff there issome p∈Pwith a,b∈ p.
Definition 4.13. Let A be a set and R⊆A× A a binary relation on A.
�etransitiveclosure of R istherelation
TC(R)∶=⋂{S⊆ A×A∣ S⊇ R istransitive}.
Since the family of transitive relations is closed under intersections
we canuse Lemma 4.8(b)toprovethat TC is a closure operator.
Lemma4.14. Let Abeaclass. TC isaclosure operator on A×A.
Exercise 4.4. Prove Lemma 4.14.
Lemma 4.15. If R⊆A×Aisasymmetricrelationthenso is TC(R).
Proof. LetS∶= TC(R)∩(TC(R)) −1 .SinceRissymmetricwe haveR⊆S.
We claimthat S istransitive.
Let⟨a,b⟩,⟨b,c⟩∈ S. �en⟨a,b⟩,⟨b,c⟩∈ TC(R) and⟨b,a⟩,⟨c,b⟩∈
TC(R). �erefore, we have⟨a,c⟩∈TC(R) and⟨c,a⟩∈TC(R). �is
impliesthat⟨a,c⟩∈ S, as desired.
We have shown that S is a transitive relation containing R. By the
definition of TC it follows that TC(R)⊆ S=TC(R)∩TC(R) −1 . �is
impliesthat TC(R) −1 = TC(R). Hence, TC(R) issymmetric. ◻
Lemma4.16. Let R⊆A×Abeabinaryrelation.
(a) �esmallestreflexiverelationcontainingRis R∪idA.
55

a2. Relations
(b) �esmallestsymmetricrelationcontainingRis R∪R −1 .
(c) �esmallesttransitiverelationcontaining R is TC(R).
(d) �esmallestequivalencerelationcontainingRisTC(R∪R −1 ∪idA).
Proof. (a) R∪idA is obviously reflexive and it contains R. Conversely,
supposethat S⊇Risreflexive. �en idA⊆ S impliesthat R∪ idA⊆S.
(b) isproved analogously.
(c) Let S⊇R be transitive. �en the intersection in the definition
of TC contains S. Hence, TC(R)⊆ S. Furthermore, we have R⊆ TC(R)
by definition. Itremainstoprovethat TC(R) istransitive.
Let⟨a,b⟩,⟨b,c⟩∈ TC(R). �en we have⟨a,b⟩,⟨b,c⟩∈ S, for every
transitive relation S⊇R. Hence, we have⟨a,c⟩∈ S, for each such relation
S.�is impliesthat⟨a,c⟩∈ TC(R).
(d)SetE∶= TC(R∪R −1 ∪ idA). Clearly,we haveR⊆Eand, ifS⊇R is
an equivalencerelationthenE⊆ S. Hence, it isremainstoprovethatE is
an equivalencerelation. It istransitive by(c),symmetric by Lemma 4.15,
and E isreflexivesince idA⊆ TC(R∪R −1 ∪ idA). ◻
Corollary4.17. LetAbeasetand F⊆℘(A×A)theset ofallequivalence
relations on A. �en⟨F,⊆⟩ forms a complete partial order. If X⊆F is
nonemptythenwe have
inf X=⋂X and supX= TC(⋃X).
Proof. By Lemma 4.16,we have F= fixcwherec isthe closure operator
with
c(R)∶= TC(R∪R −1 ∪ idA).
�e relation E∶=⋃X is reflexive and symmetric since X is nonempty.
Hence, we have TC(E∪E −1 ∪ idA)=TC(E). Consequently, the claim
follows from�eorem 4.9. ◻
56

a3. Ordinals
1. Well-orders
When definingstageswe frequentlyusedthe factthat any class ofstages
has a minimal element. In this section we study arbitrary orders with
thisproperty.
Definition1.1. Let⟨A,R⟩ be a graph.
(a) An element a∈A isR-minimal if⟨b,a⟩∈ R impliesb=a.
(b) ArelationRis le�-narrow ifR −1 (a) is aset, for everyseta∈rngR.
(c) R iswell-founded if every nonempty subset B⊆A contains an Rminimal
element. A le�-narrow, well-founded linear order is called a
well-order.
Example. (a)⟨N,≤⟩ is awell-order.
(b)⟨N,∣⟩ is awell-foundedpartial order.
(c)�e membershiprelation∈is awell-foundedpartial order onS. It
is awell-order onthe class of allstages.
(d)⟨℘(N),⊆⟩ is notwell-founded.
(e) Apartial order⟨A,≤⟩ is le�-narrow if, and only if,⇓a is aset, for
all a∈A.
Exercise1.1. Provethat⟨℘(N),⊆⟩ is notwell-founded.
Lemma1.2. If⟨A,R⟩ isawell-founded graphandB⊆Athen⟨B,R∣B⟩ is
alsowell-founded.
Proof. Every nonempty subset C⊆B is also a nonempty subset of A
and has an R-minimal element. ◻
logic, algebra & geometry2012-08-08 — ©achim blumensath 57

a3. Ordinals
Lemma 1.3. If ⟨A,≤⟩ is a well-founded and le�-narrow partial order,
there exists no infinite sequence(an)n∈N∈A N such that an≠an+1 and
an+1≤ an, forall n.
Proof. If there exists such an infinite sequence then the class rngā=
{an∣ n∈N} is nonempty and has no≤-minimal element. Furthermore,
rngā⊆⇓a0 is asetsincethe order is le�-narrow. ◻
�ereasonwhywell-foundedrelations are of interest isthatthese are
exactly thoserelations that admitproofs by induction. Asthe theorem
belowshowswe canprovethat every element of awell-foundedpartial
order⟨A,≤⟩ satisfies a given property φ by showing that, if every elementb

1. Well-orders
Example. Consider the well-order⟨N,

a3. Ordinals
Proof. Suppose that there exists some a∈Awith a> f(a). Let a0 be
the minimalsuch element. By minimality of a0 we have
f(a0)≤ f( f(a0)).
Onthe other hand,since f isstrictly increasing we have
f( f(a0))< f(a0).
Contradiction. ◻
Lemma 1.8. Let⟨A,≤⟩ be a well-order and I⊆A. �e following statementsareequivalent:
(1) I isaproper initialsegment ofA.
(2) I=↓Aa, forsome a∈A.
(3) I isan initialsegment ofAand I is non-isomorphicto A.
Proof. (1)⇒(2) If I is a proper subclass of A then A∖ I is nonempty
and has a least element a. Consequently, we have I=↓a.
(2)⇒(3) Let I=↓a. Supposethere exists an isomorphism f∶ A→ I.
By Lemma 1.7, we have f(a)≥ a. Hence, f(a)∉ I=rng f. Contradiction.
(3)⇒(1) is trivial. ◻
Corollary1.9. < isastrictpartial order onWo.
Proof. We can see immediately from the definition that

1. Well-orders
Proof. Let f , g∶A→ B be isomorphisms.�enso is g○ f −1 ∶ B→B. In
particular, g○ f −1 isstrictly increasing. By Lemma 1.7,we obtain
f(a)≤(g○ f −1 )( f(a))= g(a) , for all a∈A.
Similarly, we derive g(a)≤ f(a), for all a. It followsthat f= g. ◻
Westill have to provethat

a3. Ordinals
�erestriction of h1 to↓a0 is an isomorphism
h1↾↓a0∶↓a0→↓h1(a0).
Composing itwith h −1
0 yields an isomorphism
(h1↾↓a0)○ h −1
0 ∶↓b0→↓h1(a0).
Butthis contradicts h1(a0)

1. Well-orders
Proof. (⇒) By definition, every continuous function satisfies (2). Furthermore,
a + =sup{a,a + } impliesthat f(a + )=sup{f(a), f(a + )}.
(⇐) Forthe other direction,supposethat f satisfies(1) and(2). First,
we show that f is increasing. Suppose otherwise and let a∈A be the
minimal element such that f(b)> f(a), for some b

a3. Ordinals
Proof. Ifais the greatest element ofA,we can setb∶= a. Otherwise,we
have f(a + )> f(a)≥ a, by Lemma 1.7. Hence, there are elementsx∈A
with f(x)>a. Let c be the least such element. We have c>� since
f(c)> a≥ f(�). If c were a limitthen, by choice of c,wewould have
f(c)=sup{ f(x)∣ x< c}≤ a< f(c).
A contradiction. Hence, c is a successor and there exists some b∈A
with c=b + . By choice of c, we have f(b)≤a. Furthermore, if x>b
then x≥c, which implies that f(x)≥ f(c)> a. �erefore, b is the
desired element. ◻
2. Ordinals
We have seen that there exists a well-order on Wo if one does not distinguish
between isomorphic orders.Wewould liketo define asubclass
On⊆Wo of ordinals such that, for each well-order A, there exists a
unique element B∈Onthat is isomorphicto A.
Wewillpresenttwo approachesto doso.�eusual one – duetovon
Neumann – hasthe disadvantage that itrequiresthe Axiom ofReplacement.
Without it we cannot prove that, for every well-order α, there
exists an isomorphicvon Neumann ordinal.�erefore,wewill adopt a
different approach.�erelation≅ forms a congruence(seeSection b1.4
below) on the class of all well-orders. A first try might thus consist in
representing awell-ordering by its congruence class.Unfortunately,the
class of all well-orders isomorphic to a given one is not a set. Hence,
withthis definition one could not formsets of ordinals. Instead of consideringall
isomorphicwell-orderswewilltherefore onlytakesome of
them.
Definition 2.1. �e ordertype of awell-order A istheset
ord(A)∶=[A]≅= cut{B∣Bis awell-order isomorphicto A}.
�e elements of On∶=rng(ord) are called ordinals.
64

2. Ordinals
Instead of asubclass On⊆Wothe above definition results in a function
ord∶Wo→On. Belowwewillseethatthere exists a canonicalway
to associate with every ordinal α∈ On a well-order f(α)∈Wo. Using
this injection f∶ On→Wowe can identify the class Onwith its image
f[On]⊆Wo.
First, let us show that the mapping ord∶Wo→On has the desired
property of characterising awell-orderupto isomorphism.
Lemma 2.2. Let A and B be well-orders that are sets. �ere exists an
isomorphism f∶ A→B if,and only if, ord(A)=ord(B).
Proof. If f ∶ A → B is an isomorphism then a well-order C is isomorphicto
A if, and only if, it is isomorphicto B. �erefore ord(A)=
ord(B). Conversely, suppose ord(A)=ord(B).Since A is awell-order
isomorphic to A, we have ord(A)≠∅. Fix an arbitrary element C∈
ord(A). By definition, C is isomorphic to A and to B. Consequently,
A and B are isomorphic. ◻
Remark. Wewillprove in Lemma a4.5.3with the help of the Axiom of
Replacement that any two well-ordered proper classes are isomorphic.
In particular, it followsthat inthe above lemmawe can droptherequirement
of A and B beingsets.
Definition 2.3. Let On∶=⟨On,

a3. Ordinals
�eorem 2.4. On isawell-order.
�e notions of a successor ordinal and a limit ordinal are defined in
thesameway as for arbitrarywell-orders.Recallthatwe denotethesuccessor
of α by α + . Furthermore,we define
0∶= ord⟨∅,∅⟩ , 1∶=0 + , 2∶= 1 + ,...
�e first limit ordinal is ω∶= ord⟨N,≤⟩.
Lemma 2.5. Let α,β∈On. Ifα≤βthen S(α)⊆ S(β).
Proof. If α=β, the claim is trivial. �erefore, we assume that α

Lemma 2.7. On is notaset.
2. Ordinals
Proof. Suppose that On is a set. Since On is well-ordered there exists
some ordinal α ∈ On with α = ord⟨On,≤⟩. We have just seen that
ord⟨↓α,≤⟩=α. �erefore, there exists an isomorphism f ∶↓α→On.
But↓α is a proper initial segment of On. �is contradicts Lemma 1.8.
◻
Lemma 2.8. A subclass X⊆ On is a set if, and only if, it has an upper
bound.
Proof. (⇐) If X⊆ On has anupper bound α then X⊆⇓α.Since⇓α is
asetthe claim follows.
(⇒) Suppose that X is a set. Since On is a proper class there exists
some ordinal α∈ On∖S(X).We claim that α is anupper bound of X.
Suppose there exists some β∈X with β≰α. �en α

a3. Ordinals
Proof. (⇒) If A≤B then, by definition, there exists an isomorphism
f ∶ A→ I between A and an initial segment I of B. In particular, f ∶
A→ B is astrictly increasing function.
(⇐) Suppose that f ∶ A→ B is a strictly increasing function and
letC∶=rng f.SinceC⊆B iswell-orderedthere exists an isomorphism
g∶C→ I⊆ On betweenCand an initialsegment of On.Similarly,there
issome isomorphism h∶B→J⊆ On.We claimthat
k∶= h −1 ○ g○ f∶ A→ B
isthe desired isomorphism betweenA and an initialsegment ofB.Since
f , g, and h −1 are isomorphismsso is k.Whatremainsto beshown isthat
k is in factwell-defined,that is, I=rng g⊆rng h= J.
We claim that g(c)≤ h(c), for all c∈C. Since I and J are initial
segmentsthis impliesthat I⊆J. For a contradiction, supposethatthere
issome c∈Cwith g(c)> h(c) and let c bethe minimal such element.
Notethat, since g and h are strictly increasing and rng g and rng h are
initial segmentswe must have
g(c)= min(I∖rng(g↾↓Cc))
and h(c)=min(J∖rng(h↾↓Bc)).
By choice ofc,we haverng(g↾↓Cc)⊆rng(h↾↓Bc). But, bythe above
equations, this impliesthat g(c)≤ h(c). A contradiction. ◻
In order tousethetheory of ordinals forproofs about arbitrary sets
oneusually needsto define awell-order on a givenset. In generalthis is
only possible if one assumes the Axiom of Choice. Until we introduce
this axiomthe followingtheoremwillserve as astopgap. Oncewe have
defined the cardinality of a set in Section a4.2 it will turn out that the
ordinalthetheoremtalks about is α=∣A∣ + .
�eorem 2.12(Hartogs). For every set Athere exists an ordinal α such
thatthereare no injective functions↓α→A.
68

2. Ordinals
Proof. For a contradiction, suppose that there exists a set A such that,
for every ordinalα,there is an injective function fα∶↓α→A. LetAα∶=
rng fα⊆ A andset
Rα∶={⟨a,b⟩∈ Aα×Aα∣ f −1
α (a)≤ f−1 α (b)}.
By construction, fα∶⟨↓α,≤⟩→⟨Aα,Rα⟩ is an isomorphism. Hence, by
the definition of an ordinal,we have
S(α)⊆ S(⟨Aα,Rα⟩).
Since Rα⊆ A×A∈℘ 3 (A)⊆℘ 3 (S(A)) it followsthat
⟨Aα,Rα⟩={{Aα},{Aα,Rα}}⊆℘ 4 (S(A)).
We haveshownthat
α⊆S(α)⊆ S(⟨Aα,Rα⟩)⊆℘ 4 (S(A)) , for all α∈ On.
Consequently, On⊆℘ 5 (S(A)),which impliesthat On is aset.�is contradicts
Lemma2.7. ◻
VonNeumannordinals
We concludethissectionwith an alternative definition of ordinals.�is
definition issimpler and the resulting ordinals have many niceproperties
such that α=↓α and supX=⋃X. �e only disadvantage is that
one needs an additional axiom in order to prove that every well-order
is isomorphictosome ordinal. Intuitively, we define avon Neumann ordinalto
bethe set of all smaller ordinals, that is, α∶=↓α. Asusual, the
actual definition is moretechnical andwe havetoverify a�erwardsthat
it hasthe desired effect.
Definition 2.13. Aset α is avon Neumann ordinal if it istransitive and
linearly ordered by the membership relation∈. We denote the class of
all von Neumann ordinals by On0 andweset On0∶=⟨On0,∈⟩.
69

a3. Ordinals
Example. �e set[n]={[0],... ,[n−1]} is a von Neumann ordinal,
for each n∈N.
Lemma 2.14. Ifα∈ On0 and β∈αthen β∈On0.
Proof. First, notethat β∈α implies β⊆α. Asαis linearly ordered by∈
ittherefore followsthatso is β⊆α.
It remains to prove that β is transitive. Suppose that η∈γ∈β. By
transitivity of α,we have η, γ,β∈α.Since α is linearly ordered by∈we
know thatthe relation∈,restricted to α, istransitive. Hence, η∈γand
γ∈β impliesthat η∈β. ◻
Remark. Notethat, for α∈ On0,we have
↓α={β∈On0∣ β∈α}.
Hence, α=↓α and our definition of avon Neumann ordinal coincides
withthe intuitive one.
Exercise 2.1. Suppose that α={β0,... ,βn−1} is a von Neumann ordinalwith
n

Lemma 2.16. On0 is notaset.
2. Ordinals
Proof. On0 istransitive andwell-ordered by∈. If itwere aset, itwould
be an element of itself. ◻
On0 is linearly ordered by∈.�e followingsequence of lemmas contains
several characterisations of this ordering. In particular, we show
thatthe mapping
ord∶⟨On0,∈⟩→⟨On,

a3. Ordinals
Proof. (1)⇔(2)was already shown in Lemma 2.17.
(1)⇒(3) a∈bimplies S(a)∈ S(b), for arbitrary sets a andb.
(3)⇒(1) If α ∉ β then, by Lemma 2.15, we either have α = β or
β∈α. Consequently, either S(α)= S(β) orS(β)∈ S(α). It followsthat
S(α)∉ S(β).
(2)⇒(4) If α⊆β, the identity idα∶ α→α⊆β is an isomorphism
fromα to an initial segment of β. Hence,α

2. Ordinals
�ereason whythere might be lessvon Neumann ordinals than elements
of On is that each von Neumann ordinal is contained in a new
stage.�at is,we have exactly onevon Neumann ordinal for everystage.
Lemma 2.22. �e function f∶ On0→ H(S)definedby f(α)∶= S(α) is
an isomorphismbetween On0 andtheclass ofallstages.
Proof. By Lemma2.19 it followsthat f is injective and increasing. Suppose
that it is not surjective. Let S be the minimal stage such that S∉
rng f , andset
X∶={α∈ On0∣ S(α)∈ S}.
Since X ⊆ S, X is a set and, hence, a proper initial segment of On0.
�erefore,there issome α∈On0 suchthat X=↓α.Since S(β)∈ S, for
all β∈α, it follows that S(α)⊆S. By choice of S, we have S(α)≠ S.
Hence, S(α)∈ S,which impliesthat α∈X=↓α. Contradiction. ◻
Definition 2.23. For α∈ On0,weset Sα∶= S(α).
Remark. In On0 we have finally found the indices to enumerate the
cumulative hierarchy
S0⊂ S1⊂⋯⊂ Sα⊂ Sα+1⊂⋯
�e class of allstages can bewritten inthe form
H(S)={Sα∣ α∈On0} ,
andwe haveS=⋃{Sα∣ α∈ On0}.
Definition 2.24. �erankρ(a) of asetaisthevon Neumann ordinalα
suchthat S(a)= Sα.
Remark. (a) For α∈ On0,we have ρ(α)= α.
(b) Notethat
cutA={x∈ A∣ ρ(x)≤ ρ(y) for all y∈A}.
Lemma 2.25. AclassXisaset if,and only if,{ρ(x)∣ x∈X}isbounded.
Exercise 2.3. Provethepreceding lemma.
73

a3. Ordinals
3. Inductionand fixedpoints
�e importance of ordinals stems from the fact that they allow proofs
and constructions by induction.�e nexttheorem follows immediately
from�eorem 1.5.
�eorem 3.1(Principle ofTransfinite Induction). Let I⊆ Onbe an initialsegment
of On. If X⊆Iisaclasssuchthat, forevery α∈I,
then X=I.
↓α⊆X implies α∈X
Usually one applies this theorem in the following way. If one wants
to prove that all ordinals satisfy a certain property φ, it is sufficient to
provethat
◆ 0satisfies φ;
◆ if α satisfies φthenso doesα + ;
◆ if δ is a limit ordinal and every α

3. Inductionand fixedpoints
If β

a3. Ordinals
which impliesthat
F↾↓α= fβ↾↓α , for all β≥α.
�erefore, it followsthat
F(α)= fα +(α)= H( fα +↾↓α)=H(F↾↓α).
In particular, if F is a set then F= fα, for some α. Hence, we have
dom F= dom fα =↓α. Since α∉dom F it follows that fα + does not
exists. Hence, H( fα)=H(F) is undefined and F∉ dom H. If F is a
proper classthenwetrivially have F∉ dom H. ◻
Remark. A�er we have introduced the Axiom of Replacement we can
actually showthat, if H∶S

y
α+0 ∶= α ,
α+β + ∶=(α+β) + ,
3. Inductionand fixedpoints
α+δ ∶=sup{α+β∣β

a3. Ordinals
Remark. (a) Notethat, if A is asetthen, bythePrinciple ofTransfinite
Recursion, there exists a unique function F∶ On→A satisfying the
above equations provided we can show that, for every limit δ, the supremum
sup F[↓δ] exists. If, furthermore, we can prove that F(β + )≥
F(β), for all β,then it followsthat f is inductive.
(b) Every fixed-point induction F is continuous, by Lemma 1.13.
Example. (a) �e function f∶ On→On∶ α↦α + is inductive. Its fixedpoint
induction over0isthe identity function F∶ On→On∶ α↦α.
(b) Let f∶S→S bethe functionwith f(a)∶=℘(a).�e fixed-point
induction of f over∅isthe function F∶ On0→Swith
F(α)∶= Sα.
(c)�e graph of addition
A∶={(x,y,z)∈ N 3 ∣ x+y=z}
isthe least fixed point ofthe function f∶℘(N 3 )→℘(N 3 )with
f(R)∶={(x,0,x)∣ x∈N}
∪{(x,y+ 1,z+ 1)∣(x,y,z)∈ R}.
Its fixed-point induction over∅isthe function
⎧⎪{(x,y,z)∣
x+y=z ,y

and, generally, we have
F(n)= ⋃k

a3. Ordinals
Proof. F(α) is a fixed point of f since f(F(α))= F(α + )= F(α). ◻
�us,we canusethe fixedpoint induction F of f to compute a fixed
pointprovided F converges.
Lemma 3.8. Let F bethe fixed-point induction ofafunction f. If F(α)=
F(α + )then F(α)= F(β), forall β≥α.
Proof. Weprovethe claim by induction on β. If β=αthenthe claim is
trivial. Forthesuccessorstep,we have
F(β + )= f(F(β))= f(F(α))= F(α + )= F(α).
Finally, ifδ> α is a limit ordinal,then
F(δ)=sup{ F(β)∣ β

3. Inductionand fixedpoints
Definition 3.10. Let f ∶ A→ A be inductive and F ∶ On→A the
corresponding fixed-point induction. �e minimal ordinal α suchthat
F(α)=F(α + ) is called the closure ordinal of the induction and the
element F(∞)∶= F(α) isthe inductive fixedpoint of f over a.
Remark. IfA is aset, every inductive function f∶ A→ A has an inductive
fixedpoint.
Example. Let⟨A,R⟩ be a graph.�ewell-foundedpart of R isthe maximalsubsetB⊆Asuchthat⟨B,R∣B⟩
iswell-founded and, for all⟨a,b⟩∈
R with b∈B,we also have a∈B.We can compute B as inductive fixed
point over∅ofthe function
f(X)∶={x∈ A∣ R −1 (x)⊆ X∪{x}}.
Ifwewantto applythe above machineryto compute fixedpoints,we
need methods to show that a given function f is inductive. Basically,
there are two conditions a function f has to satisfy. �e sequence obtained
by iterating f hasto be linearly ordered and itssupremum must
exists.
Definition 3.11. Let A=⟨A,≤⟩ be apartial order.
(a) A is inductively ordered if every chain C⊆A that is a set has a
supremum.
(b) A function f∶ A→ A is inflationary if f(a)≥ a, for all a∈A.
Remark. (a) Every inductively ordered set has a least element�since
theset∅is linearly ordered.
(b) Every completepartial order is inductively ordered.
(c)⟨On,≤⟩ is inductively ordered.
(d) If⟨A,≤⟩ is a well-order then according to Lemma 1.7 all strictly
continuous functions f∶ A→ A are inflationary.
Example. (a)�epartial order⟨F,⊆⟩where
F∶={X⊆N∣ X is finite}
81

a3. Ordinals
is not inductively orderedsincethe chain
[0]⊂[1]⊂[2]⊂⋅⋅⋅⊂[n]⊂⋯
has noupper bound.
(b) LetV be avector space overthe field K andset
I∶={B⊆V∣ B is linearly independent}.
We claimthat⟨I,⊆⟩ is inductively ordered.
LetC⊆I be a chain.WeshowthatsupC=⋃C. By Corollary a2.3.10,
it issufficienttoprovethat⋃C∈I.
Suppose otherwise. �en⋃C is not linearly independent and there
are elements v0,... ,vn∈⋃C and λ0,... , λn∈ K suchthat λi≠ 0, for
all i, and
λ0v0+⋅⋅⋅+ λnvn=0.
For eachvi, fixsome Bi∈ C withvi∈ Bi.SinceCis linearly orderedso
istheset{B0,... ,Bn}.�isset is finite and,therefore, it has a maximal
element Bk, that is, Bi ⊆ Bk, for all i. It follows that v0,... ,vn∈ Bk,
which impliesthat Bk is not linearly independent. Contradiction.
Lemma 3.12. Let A=⟨A,≤⟩be inductively ordered.
(a) If f∶ A→ A is inflationary, f is inductive overeveryelementa∈A.
(b) If f∶ A→ A is increasing, f is inductive overeveryelement awith
f(a)≥ a.
(c) If f∶ A→ A iscontinuous, f is inductive overeveryelementawith
f(a)≥ a. Furthermore, ifthe inductive fixed point of f over a exists, its
closure ordinal isat most ω.
Proof. (a) Bytransfiniterecursion,we construct an increasing function
F ∶ I → A satisfying the equations in Definition 3.6. Let F(0)∶= a.
For the induction step, suppose that F(α) is already defined. We set
F(α + ) ∶= f(F(α)). Since f is inflationary, it follows that F(α + ) =
f(F(α))≥F(α). Finally, suppose that δ is a limit ordinal. If F↾↓δ
82

3. Inductionand fixedpoints
is aproper class, we are done. Otherwise, F[↓δ] is asetwhich, furthermore,
is linearly ordered because F↾↓δ is increasing. As⟨A,≤⟩ is inductively
ordered it followsthat F[↓δ] has asupremum andwe canset
F(δ)∶=sup F[↓δ].
(b) Again we define an increasing function F∶ I→A by transfinite
recursion. Let F(0)∶= a. For the induction step, suppose that F(α) is
already defined.Weset F(α + )∶= f(F(α)).Toprovethat F(α + )≥ F(α)
we consider three cases. For α= 0we have F(1)= f(a)≥ a=F(0). If
α=β + is a successor, we know by induction hypothesis that F(β + )≥
F(β).Since f is increasing it followsthat
F(α + )= f(F(β + ))≥ f(F(β))= F(β + )= F(α).
If α is a limitthen F(α)=sup F[↓α] and
F(α + )= f(sup F[↓α])≥ f(F(β))= F(β + ) , for all β

a3. Ordinals
(a) f is inductive overα.
(b) If F is the fixed-point induction of f over α then F(∞) exists if,
and only if, the set{f n (α)∣ n

4. Ordinalarithmetic
�esecondtheorem is aversion ofthe�eorem of Knaster andTarski
whichshowsthat we can computethe least fixed point of a function f
by a fixed-point induction.
�eorem 3.15. Let⟨A,≤⟩bean inductively ordered graphwhereA isaset
and let f∶ A→ Abe an increasing function. If the least fixed point of f
exists then itcoincideswith its inductive fixedpoint over�.
Proof. Let F∶ On→ A bethe fixed-point induction of f over�.Suppose
thata∶= lfp f exists.Weprove by induction onαthat F(α)≤ a.�en it
followsthat F(∞)≤ a andthe minimality of a impliesthat F(∞)= a.
Clearly, F(0)=�≤ a. Forthe inductive step,supposethat F(α)≤ a.
Since f is increasing it followsthat
F(α + )= f(F(α))≤ f(a)= a.
Finally, if δ is a limit ordinal,the induction hypothesis impliesthat
F(δ)=sup{ F(α)∣ α

a3. Ordinals
+ = ⋅ =
andthe order is defined by
Figure 1..Sum andproduct of linear orders
⟨i,a⟩≤C⟨k,b⟩ : iff i=k=0and a≤A b
or i=k= 1 and a≤B b
or i=0and k= 1.
(b)�eproduct A⋅Bisthe graph⟨C,≤C⟩whereC∶= A×B andthe
order is defined by
⟨a,b⟩≤C⟨a ′ ,b ′ ⟩ : iff b

(c) K (M) ≅⟨[k m ],≤⟩.
4. Ordinalarithmetic
Addition of linear orders is associative andthe empty order is a neutral
element. Belowwe will give an example showing that, in general, it
is not commutative.
Lemma4.2. If A, B,and Care linear ordersthen
(A+B)+C≅A+(B+C).
Proof. Let A=⟨A,≤A⟩, B=⟨B,≤B⟩, and C=⟨C,≤C⟩. We can define a
bijection f∶(A⊍B)⊍C→A⊍(B⊍C) by
f⟨0,⟨0,a⟩⟩∶=⟨0,a⟩ for a∈A ,
f⟨0,⟨1,b⟩⟩∶=⟨1,⟨0,b⟩⟩ forb∈B ,
f⟨1,c⟩∶=⟨1,⟨1,c⟩⟩ for c∈C.
Sincethis bijectionpreservesthe ordering it isthe desired isomorphism.
◻
As we want to define arithmetic operations on ordinals we have to
showthat, ifwe applythe above operationstowell-orders,we again obtain
awell-order.
Lemma 4.3. If A and B are well-orders then so are A+B, A⋅B, and
A (B) .
Proof. Suppose that A=⟨A,≤A⟩ and B=⟨B,≤B⟩. We will prove the
claim only for C∶= A (B) .�e other operations are le� as an exerciseto
thereader.
Let C=⟨C,≤C⟩. �e relation

a3. Ordinals
that, respectively, f(b0)≠ g(b0) and g(b1)≠ h(b1). By definition, we
have f(b0)

is a bijection sincewe have
g(c)= g ′ (c) for all g, g ′ ∈ Z and every c≥b.
4. Ordinalarithmetic
Furthermore, ρ preserves the ordering, that is, it is an isomorphism. It
followsthat ρ −1 (h) isthe minimal element of Z and of X. ◻
Exercise 4.2. Show that, if A and B are well-orders then so are A+B
and A⋅B.
It is easy to see that A ≅ A ′ and B ≅ B ′ implies that the sums,
products, andpowers are also isomorphic.�erefore,we can definethe
corresponding operations on ordinals bytaking representatives.
Definition 4.4. For α= ord(A) and β=ord(B)we define
α+β∶= ord(A+B) ,
α⋅β∶= ord(A⋅B) ,
α (β) ∶= ord(A (B) ).
Example. �e following equations can beproved easily bythe lemmas
below.We encouragethereaderto derivethem directly fromthe definitions.
1+1=2 (3+6)ω= 9ω=ω

a3. Ordinals
Ordinaladdition
�e properties of ordinal addition, multiplication, and exponentiation
aresimilarto, but notquitethesame asthose for integers.�e following
sequence of lemmassummarisesthem.Westartwith addition.
Lemma 4.5. Let α,β, γ∈On. If β

4. Ordinalarithmetic
For a contradiction suppose that γ

a3. Ordinals
�e next lemmasummarisesthe laws of ordinal addition.
Lemma 4.9. Let α,β, γ∈On.
(a) α+(β+γ)=(α+β)+γ.
(b) α+β=α+ γ implies β=γ.
(c) α≤β implies α+ γ≤β+γ.
(d) If X⊆ On is nonemptyandboundedthen
α+supX=sup{α+β∣β∈X}.
(e) β≤αif,and only if, α=β+ γ, forsome γ∈ On.
(f) β

4. Ordinalarithmetic
(e) If β

a3. Ordinals
Proof. Fix representatives α= ord(A) and β=ord(B).
(a) follows immediately from the fact that A⋅⟨∅,∅⟩=⟨∅,∅⟩.
(b) �e canonical bijection
given by
A×(B⊍[1])→(A×B)⊍A
⟨a,⟨0,b⟩⟩↦⟨0,⟨a,b⟩⟩ ,
⟨a,⟨1,0⟩⟩↦⟨1,a⟩ ,
induces an isomorphism
A⋅(B+⟨[1],≤⟩)→A⋅B+A.
(c) Let X∶={αβ∣β

4. Ordinalarithmetic
We can also show that ordinals allow a limited form of division.
Lemma 4.13. For all ordinals α,β∈On with β≠0, there exist unique
ordinals γand ρα=βγ+ρ,
which impliesthat ρβδ+σ= α.
A contradiction. It followsthat γ isunique. Hence,theuniqueness of ρ
follows from Lemma 4.8. ◻
Lemma4.14. α isalimit ordinal if,and only if, α=ωβ, forsome β>0.
Proof. (⇒) By Lemma 4.13,we have α=ωβ+n for some β∈On and
n

a3. Ordinals
Lemma4.15. Let α,β, γ∈On.
(a) α(βγ)=(αβ)γ.
(b) α(β+ γ)= αβ+αγ.
(c) If α≠0andαβ=αγ then β= γ.
(d) α≤β implies αγ≤βγ.
(e) If X⊆ On is nonemptyandboundedthen
α⋅supX=sup{αβ∣β∈ X}.
Proof. (b)Weprovethe claim by induction on γ. For γ=0,we have
α(β+0)= αβ=αβ+0=αβ+α0.
Forthesuccessorstep,we have
α(β+ γ + )= α(β+γ) +
= α(β+γ)+α
= αβ+αγ+α
= αβ+αγ + .
Finally, if γ is a limit ordinalthen
α(β+ γ)= α⋅sup{β+ρ∣ρ

Ordinalexponentiation
4. Ordinalarithmetic
Finally, we consider ordinal exponentiation. Again, the basic steps are
thesame as for addition and multiplication.
Lemma4.16. Let α,β, γ∈On. Ifα> 1and β

a3. Ordinals
For a contradiction suppose that γ

4. Ordinalarithmetic
Proof. By Corollary 4.18 and Lemma 1.14,there exists a greatest ordinal η
suchthat β (η) ≤ α, and, by Lemma 4.13, there exist ordinals γ and ρ<
β (η) suchthat β (η) γ+ρ=α. If γ= 0,wewould have ρ=α≥β (η) > ρ.
A contradiction. And, if γ≥β,wewould have
α 1thenwe have β

a3. Ordinals
Cantor normal form
We can applythe logarithm to decompose every ordinal in a canonical
way.
�eorem 4.21. For all ordinals α,β∈On with β>1, there are unique
finitesequences(γi)iηn−1 , and 0ρ1>... of ordinals which is impossible. Consequently,
there issome number n suchthat ρn=0 andwe have
α=β (η0) γ0+⋯+β (ηn−1) γn−1. ◻
Definition 4.22. Letαbe an ordinal.�eunique decomposition
α=ω (η0) γ0+⋯+ω (ηn) γn ,
with η0>⋯>ηn and 0

Proof. Supposethat β=α+ γ, for γ>0.We have
ω (α) + ω (β) = ω (α) + ω (α+γ)
= ω (α) + ω (α) ω (γ)
= ω (α) (1+ ω (γ) )
= ω (α) ω (γ)
4. Ordinalarithmetic
= ω (α+γ) = ω (β) . ◻
Corollary 4.24. Let α,β∈Onbe ordinalswithCantor normal form
α=ω (η0) k0+⋯+ ω (ηm−1) km−1 ,
β=ω (γ0) l0+⋯+ω (γn−1) ln−1.
If i isthe maximal indexsuchthat ηi≥ γ0 thenwe have
α+β=ω (η0) k0+⋯+ ω (ηi) ki+ ω (γ0) l0+⋯+ω (γn−1) ln−1.
Lemma 4.25. An ordinal α> 0 is of the form α=ω (η) , for some η, if,
and only if, β+ γ

a3. Ordinals
If k> 1,weset β∶= ω (η) (k− 1)+ρ0. Inthis casewe cansetβ∶= ω (η) andwe
have
β+β=ω (η) + ω (η) > ω (η) +ρ=α.
Again a contradiction. ◻
�e nexttwo lemmasprovidethe laws of multiplication and exponentiation
of ordinals in Cantor normal form.
Lemma 4.26. If γ>0,0≤ ρ

Example. Bythe above lemmaswe have
(ω (ω(5) +ω4+2) + ω (5) ) (ω (2) 2+ω+1)
=(ω (ω(5) +ω4+2) + ω (5) ) (ω (2) 2) ⋅(ω (ω (5) +ω4+2) + ω (5) ) (ω) ⋅
⋅(ω (ω(5) +ω4+2) + ω (5) )
4. Ordinalarithmetic
=(ω ((ω(5) +ω4+2)ω (2) ) ) (2) ⋅ ω ((ω (5) +ω4+2)ω) ⋅(ω (ω (5) +ω4+2) + ω (5) )
=(ω (ω(7) ) ) (2) ⋅ ω (ω (6) ) ⋅(ω (ω (5) +ω4+2) + ω (5) )
= ω (ω(7) 2) ⋅ ω (ω (6) ) ⋅(ω (ω (5) +ω4+2) + ω (5) )
= ω (ω(7) 2+ω (6) ) ⋅(ω (ω (5) +ω4+2) + ω (5) )
= ω (ω(7) 2+ω (6) ) ⋅ ω (ω (5) +ω4+2) + ω (ω (7) 2+ω (6) ) ⋅ ω (5)
= ω (ω(7) 2+ω (6) +ω (5) +ω4+2) + ω (ω (7) 2+ω (6) +5) .
Exercise 4.5. Computethe cantor normal form of
(ω (ω(2) 7+ω3+4) 3+ ω (ω6+3) 4+ω (4) 3+1) (ω (2) 5+ω7+2)
Remark. We will prove in Lemma a4.5.6 that we can find, for every β,
arbitrarily large ordinals α0,α1,α2 suchthat
α0=β+α0 , α1=βα1 , and α2=β (α2) .
Inparticular,there are ordinals εsuchthat ε=ω (ε) . By εα we denotethe
α-th ordinal such that β (εα) = εα, for all β

a3. Ordinals
ordinals are
0, 1, 2, 3, . . .
. . . , ω, ω+1, ω+2, . . .
. . . , ω2, ω2+1, ω2+ 2, . . .
. . . , ω3, . . . , ω4, . . . , ω (2) , . . . , ω (3) , . . .
. . . , ω (ω) , . . . , ω (ω(ω) ) , . . .
. . . , ε0, . . . , ε (ε0)
0 , . . . , ε1, . . . , ε2, . . . , εω, . . .
. . . , ω1, . . . , ω2, . . . , ωω, . . .
�e ordinals ωα will be defined in Section a4.2.
104

a4. Zermelo-Fraenkelsettheory
1. �eAxiom ofChoice
We have seen that induction is apowerfultechniquetoprovestatements
and to construct objects. But in order tousethis toolwe have torelate
the sets we are interested in to ordinals. In basic set theory this is not
alwayspossible.�erefore,wewill introduce a new axiomwhichstates
that, for everysetA,there is awell-order overA. Before doingso, letus
present several statements that are equivalent to this axiom. We need
two new notions.
Definition1.1. Aset F⊆℘(A) has finite character if, for all sets x⊆ A,
we have
x∈F iff x0∈ F , for every finitesetx0⊆ x.
Lemma1.2. Suppose that F⊆℘(A) has finitecharacter.
(a) F isan initialsegment of℘(A).
(b) If X⊆ F is nonemptythen⋂X∈F.
(c) IfC⊆ F isachainand⋃C isasetthen⋃C∈ F.
Proof. (a) follows immediately from the definition and (b) is a consequence
of (a). For (c), let C⊆Fbe a chain such that X∶=⋃C is a set.
If X0⊆X is finite, there exists some element Z∈ C with X0⊆ Z∈F.
Hence,X0∈ F, for all finitesubsetsX0⊆ X.�is impliesthatX∈ F. ◻
Lemma1.3. If F has finitecharacterthen⟨F,⊆⟩ is inductively ordered.
logic, algebra & geometry2012-08-08 — ©achim blumensath 105

a4. Zermelo-Fraenkelsettheory
Proof. LetC⊆ F be a linearly orderedsubset of F. By Corollary a2.3.10
and Lemma 1.2 (c), it follows thatsupC=⋃C∈ F. ◻
Example. LetV be avector space overthe field K.�eset
F∶={B⊆V∣ B is linearly independent}
has finite character.
�esecond notionwe need isthat of a choice function. Intuitively, a
choice function is a functionthat, given someset A,selects an element
of A.
Definition 1.4. A function f is a choice function if f(a)∈ a, for all
a∈ dom f.
Exercise1.1. LetI betheset of all open intervals(a,b) ofreal numbers
a,b∈Rwith a

1. �eAxiom ofChoice
Proof. Let f∶℘(A)∖{∅}→A be a choice function. We define a function
g∶℘(A)→℘(A) by
⎧⎪ A if X=A ,
g(X)∶= ⎨
⎪⎩
X∪{ f(A∖X)} if X≠A.
Since g(X)⊇ X this function is inflationary. Furthermore, the partial
order⟨℘(A),⊆⟩ is complete. By �eorem a3.3.14, g has an inductive
fixed point. Since g(X)≠ X, for X≠ A, it follows that this fixed point
is A. Let G ∶ On→℘(A) be the fixed-point induction of g over∅
and let α bethe closure ordinal. For every β

a4. Zermelo-Fraenkelsettheory
Proof. (2)⇒(3) If∏i∈I Ai is a proper class, it is nonempty and we are
done. Hence,we may assumethat it is aset.�enA∶=⋃{Ai∣ i∈I}is
also aset. By(2)there exists a choice function f∶℘(A)∖{∅}→ A. Let
g∶I→A bethe function defined by g(i)∶= f(Ai). Since g(i)∈ Ai it
followsthat g∈∏i∈I Ai≠∅.
(3)⇒(4) is trivial.
(4)⇒(2) Let I∶=℘(A)∖{∅} and setAX∶= X×{X}, forX∈ I. Since
∏X∈I AX≠∅ there exists some element f ∈∏X∈I AX. We can define
the desired choice function g∶℘(A)∖{∅}→ A by
g(X)= a : iff f(X)=⟨a,X⟩.
(2)⇒(1)wasproved in Lemma 1.6.
(1)⇒(5) Suppose that⟨A,≤⟩ is inductively ordered, but A has no
maximal element. For every a∈A,we can find someb∈Awith b>a.
By assumption,there exists awell-orderRover A. Let f∶ A→ A bethe
functionsuchthat f(a) isthe R-minimal elementb∈Awith b>a. By
definition, we have f(a)> a, for all a∈A. Hence, f is inflationary and,
by�eorem a3.3.14, f has a fixed point a. But f(a)= a contradicts the
definition of f.
(5)⇒(6) Let F be aset of finite character andA∈ F. It issufficientto
provethatthesubset F0∶={X∈ F∣ A⊆ X}is inductively ordered by⊆.
By Lemma 1.3, we know that⟨F,⊆⟩ is inductively ordered. Let C be a
chain in F0.�enC⊆F0⊆ F and C is also a chain in F. Consequently,
it has a least upper bound B∈F.Since A⊆ X, for all X∈C, it follows
thatA⊆ B,that is,B∈ F0 andB is alsothe leastupper bound ofC in F0.
(6)⇒(2) Let A be a set. By Lemma 1.5(a), the set C of choice functions
f with dom f ⊆℘(A)∖{∅} has finite character and, therefore,
there is a maximal element f∈ C. By Lemma 1.5(b), it followsthat f is
the desired choice function.
(1)⇒(7) Fixwell-ordersRandSon,respectively, A andB. By Corollary
a3.1.12, exactly one of the following conditions is satisfied:
108
⟨A,R⟩⟨B,S⟩.

1. �eAxiom ofChoice
In the first two cases there exists an injection A→ B and in the second
and third case there exists an injection B→A in the other direction.
(7)⇒(1) LetA be a set. By �eorem a3.2.12, there exists an ordinalα
such that there is no injective function↓α→A. Consequently, there
exists an injective function f∶ A→↓α.We define arelation R onA by
R∶={⟨a,b⟩∣ f(a)< f(b)}.
Since f is injective andrng f⊆↓α iswell-ordered it followsthatRisthe
desiredwell-order onA.
(2)⇒(8) Let h∶℘(A)∖{∅}→ A be a choice function.We can define
g∶B→A by
g(b)∶= h( f −1 (b)).
(8)⇒(4) Let(Ai)i∈I be a family of disjoint nonemptysets.We define
a function f∶⋃{Ai∣ i∈I}→ I by
f(a)= i : iff a∈Ai .
SincetheAi are disjoint and nonempty it followsthat f iswell-defined
and surjective. Hence, there exists a function g∶I→⋃{Ai∣ i∈I}
such that f(g(i))= i, for all i∈I. By definition of f , this implies that
g(i)∈ Ai. Hence, g∈∏i∈I Ai≠∅. ◻
Axiom of Choice. Foreveryset Athereexistsawell-orderRoverA.
Lemma 1.8. A le�-narrow partial order(A,≤) is well-founded if, and
only if,there exists no infinitestrictlydecreasingsequence a0>a1>....
Proof. One directionwas alreadyproved in Lemma a3.1.3. For the other
one, fix a choice function f∶℘(A)∖∅→A. Supposethatthere exists
a nonempty set A0⊆ Awithout minimal element. We can define a descending
chain a0>a1>... by induction. Let a0∶= f(A0) and, for
k>0,set
ak∶= f({b∈A0∣ b

a4. Zermelo-Fraenkelsettheory
Notethat ak iswell-defined since ak−1 is not a minimal element of A0.
◻
Exercise1.2. We call asetacountable ifthere exists a bijection↓ω→a.
Provethat a le�-narrowpartial order⟨A,≤⟩ iswell-founded if, and only
if, every countable nonemptysubset X⊆A has a minimal element.
Exercise 1.3. Let⟨A,R⟩ be a well-founded partial order that is a set.
Provethatthere exists awell-order≤onAwith R⊆≤.
�e following variant of the Axiom of Choice (statement (5) in the
abovetheorem) is known as‘Zorn’s Lemma’.
Lemma1.9(Kuratowski,Zorn). Every inductively orderedpartial order
hasamaximalelement.
Example. We haveseenthatthesystem of all linearly independentsubsets
of a vector space V is inductively ordered. It follows that every
vector space contains a maximal linearly independent subset, that is, a
basis.
�is example can be generalised to a certain kind of closure operators.
Definition1.10. Letcbe a closure operator on A.
(a) c hastheexchangeproperty if
b∈c(X∪{a})∖c(X) implies a∈c(X∪{b}).
(b) Aset I⊆A is c-independent if
a∉c(I∖{a}) , for all a∈I.
We call D⊆Ac-dependent if it is not c-independent.
(c) Let X⊆A. Aset I⊆X is a c-basis of X if I is c-independent and
c(I)= c(X).
110

1. �eAxiom ofChoice
Lemma 1.11. Let c be a closure operator on A and let F⊆℘(A) be the
class of all c-independent sets. If c has finite character then F has finite
character.
Proof. Let I∈Fand I0⊆ I. For every a∈I0,we have
a∉c(I∖{a})⊇ c(I0∖{a}).
Hence, I0 isc-independent. Conversely, supposethat I∉F.�enthere
issome a∈Iwith
a∈c(I∖{a}).
Since c has finite characterwe can find a finitesubset I0⊆ I∖{a}with
a∈c(I0).�us, I0∪{a} is a finitesubset of I that is notc-independent.
◻
Before proving the converse let usshow with the help of the Axiom
of Choicethatthere is always a c-basis.Westartwith an alternative description
ofthe exchangeproperty.
Lemma1.12. Letcbeaclosure operator onAwiththeexchangeproperty.
If D⊆A isaminimalc-dependentsetthen
a∈c(D∖{a}) , forall a∈D.
Proof. Let a∈D.Since D isc-dependentthere existssome elementb∈
D with b∈c(D∖{b}). Ifb=a thenwe are done. Hence,supposethat
b≠aand let D0∶= D∖{a,b}. By minimality of D we haveb∉c(D0).
Hence,b∈c(D0∪{a})∖c(D0) andthe exchangeproperty impliesthat
a∈c(D0∪{b}). ◻
Proposition1.13. Letcbeaclosure operator onAthat has finitecharacter
andtheexchangeproperty.Everyset X⊆A hasa c-basis.
111

a4. Zermelo-Fraenkelsettheory
Proof. �e family F of all c-independentsubsets of X has finite character.
By the Axiom of Choice, there exists a maximal c-independentset
I⊆X.We claimthat c(I)= c(X),that is, I is a c-basis of X.
Clearly, c(I)⊆ c(X). If X⊆c(I), it followsthat
c(X)⊆ c(c(I))= c(I)
and we are done. Hence, it remains to consider the case that there is
some elementa∈X∖c(I).We derive a contradictiontothe maximality
of I byshowingthat I∪{a} isc-independent.
Suppose that I∪{a} is not c-independent. Since F has finite character
there exists a finite c-dependent subset D⊆I∪{a} with a∈D.
Suppose that D is chosen minimal. By Lemma 1.12, it follows that a∈
c(D∖{a})⊆ c(I). A contradiction. ◻
Proposition 1.14. Let c be a closure operator on A with the exchange
property and let F⊆℘(A) be the class of all c-independent sets. �en
c has finitecharacter if,and only if, F has finitecharacter.
Proof. (⇒) has already beenproved in Lemma 1.11.
(⇐) For a contradiction, supposethatthere is aset X⊆Asuchthat
Z∶=⋃{c(X0)∣ X0⊆Xis finite}
is apropersubset of c(X). Fixsome element a∈c(X)∖Z. ByProposition
1.13 there exists a c-basis I for X. It follows that a∈c(X)= c(I).
Since F has finite character we can find a finite subset I0⊆Isuchthat
I0∪{a} is c-dependent. By Lemma 1.12, it follows that a∈c(I0)⊆Z.
A contradiction. ◻
A more extensive treatment of closure operators with the exchange
propertywill be given inSection f1.1.
112

2. Cardinals
2. Cardinals
�e notion ofthe cardinality of aset is avery natural one. It is based on
thesame ideawhich ledtothe definition ofthe ordertype of awell-order.
But instead ofwell-orderswe consider justsetswithout anyrelation. Although
conceptuallysimplerthan ordinalswe introduce cardinalsquite
late inthe development of ourtheorysince most oftheirproperties cannot
beprovedwithoutresortingto ordinals andthe Axiom of Choice.
Intuitively,the cardinality of asetA measures itssize,that is,the number
of its elements. So, how dowe count the elements of aset?We can
saythat‘A hasα elements’ ifthere exists an enumeration ofA of lengthα,
that is, a bijection↓α→A. For infinitesets,such an enumeration is not
unique.We can findseveralsequences↓α→Awith differentvalues ofα.
To get awell-defined numberwethereforepickthe least one.
Definition 2.1. �ecardinality∣A∣ of a classA isthe least ordinalαsuch
thatthere exists a bijection↓α→A. Ifthere exists nosuch ordinalthen
wewrite∣A∣∶=∞. Let Cn∶= rng∣⋅∣⊆On betherange ofthis mapping.
(We do not consider∞to be an element of the range.) We set Cn∶=
⟨Cn,≤⟩.�e elements of Cn are called cardinals.
Remark. Clearly, if∣A∣,∣B∣

a4. Zermelo-Fraenkelsettheory
(2) �ereexistsan injective function A→ B.
(3) �ereexistsansurjective function B→A.
Proof. Set κ∶=∣A∣ and λ∶=∣B∣ and let g∶↓κ→A and h∶↓λ→B be the
corresponding bijections.
(1)⇒(2) Since κ ≤ λ there exists an isomorphism f ∶ ↓κ → I
between↓κ and an initial segment I⊆↓λ. In particular, f is injective.
�e composition h○ f○ g −1 ∶ A→ B isthe desired injective function.
(2)⇒(1) For a contradiction, suppose that there exists an injective
functionA→ B butwe have∣A∣>∣B∣. By(1)⇒(2),the latter impliesthat
there is an injective function B→A. Hence, applying �eorem a2.1.12
we find a bijection A→ B. It followsthat∣A∣=∣B∣. Contradiction.
(2)⇒(3) Let f ∶ A→ B be injective. By Lemma a2.1.10 (b), there
exists a function g ∶ B → A such that g○ f = idA. Furthermore, it
follows by Lemma a2.1.10(d)that g issurjective.
(3)⇒(2) As above, given a surjective function f ∶ B→A we can
apply Lemma a2.1.10(and the Axiom of Choice) to obtain an injective
function g∶A→ Bwith f○ g= idB. ◻
For every cardinal, there is a canonical setwiththis cardinality.
Lemma 2.4. Forevery cardinal κ∈Cn,we have κ=∣↓κ∣. It followsthat
Cn={α∈On∣∣↓α∣= α}.
Exercise 2.1. Let α and β be ordinals suchthat∣α∣≤ β≤α.Show that
∣α∣=∣β∣.
Exercise 2.2. Prove that α∈ Cn, for every ordinal α≤ω. Hint. Show,
by induction on α, that there is no surjective function↓α→↓β with
α

2. Cardinals
Proof. By �eorem a2.1.13, there exists an injective functionA→℘(A)
but no surjective one. By Lemma 2.3, it follows that∣A∣≤∣℘(A)∣ and
∣℘(A)∣≰∣A∣. ◻
Cn is a proper class since it is anunboundedsubclass of On.
Lemma 2.6. Cn isaproperclass.
Proof. For a contradiction, suppose otherwise. By Lemma a3.2.8, it follows
that there is someα∈ Onsuchthat κ∣↓α∣,
which impliesthat λ>α. A contradiction. ◻
Lemma 2.7. On0≤ Cn≤On.
Proof. Since Cn⊆On it followsthat Cn is awell-order.�erefore,there
exists an isomorphism h∶Cn→I, forsome initial segment I⊆ On.
By �eorem 2.5 we know that the function f ∶ On0 → Cn with
f(α)∶=∣Sα∣ is strictly increasing. Consequently, we have On0 ≤ Cn,
by Lemma a3.2.11. ◻
Remark. With the Axiom of Replacement which we will introduce in
Section 5 we can actually prove that⟨On0,∈⟩≅⟨On,

a4. Zermelo-Fraenkelsettheory
Note that, by our definition of a cardinal, we have ωα = ℵα and
ℵ0= ω0= ω. Furthermore,ℵ + α=ℵα+1.Sincewe have defined the operation
κ + differently for cardinals and ordinalswewill usethis notation
only for cardinals intheremainder ofthis book. Ifwe considerthesuccessor
of an ordinal α wewillwriteα+ 1.
3. Cardinalarithmetic
Similarly to ordinals we can define arithmetic operations on cardinals.
Notethat, except for finite cardinals,these operations are different from
the ordinal operations. �erefore,we have chosen differentsymbolsto
denotethem.
Definition 3.1. Let κ, λ∈Cn be cardinals. We define
κ⊕λ∶=∣↓κ⊍↓λ∣ , κ⊗ λ∶=∣↓κ×↓λ∣ , κ λ ∶=∣↓κ ↓λ ∣.
�e following lemmas follows immediately fromthe definition if one
recalls that, for κ∶=∣A∣ and λ∶=∣B∣, there exist bijections A→↓κ and
B→↓λ.
Lemma 3.2. Let AandBbesets.
∣A⊍B∣=∣A∣⊕∣B∣ , ∣A×B∣=∣A∣⊗∣B∣ , ∣A B ∣=∣A∣ ∣B∣ .
Corollary 3.3. Forallα,β∈On,we have
∣↓(α+β)∣=∣↓α∣⊕∣↓β∣ and ∣↓(αβ)∣=∣↓α∣⊗∣↓β∣.
�e corresponding equation for ordinal exponentiation will be delayed
until Lemma 4.4.
Exercise 3.1. Prove that, if A is a set then∣℘(A)∣=2 ∣A∣ . Hint.Take the
obvious bijection℘(A)→2 A .
116
For finite cardinalsthese operations coincidewiththeusual ones.

Lemma 3.4. For m, n

a4. Zermelo-Fraenkelsettheory
(e) Ifthere is an injective function f∶ B→C,we can define an injective
function A⊍B→A⊍C by
⟨0,a⟩↦⟨0,a⟩ and ⟨1,b⟩↦⟨1, f(b)⟩.
(f) If κ≥ℵ0= ω then κ=ω+α, for some α∈On. We can define a
bijection↓ω→↓(ω+1) by0↦ω and n↦n−1, for n>0.�is function
can be extended to a bijection↓ω⊍↓α→↓ω⊍↓α⊍[1]. Conversely, if
κκ. ◻
Lemma 3.6. Let κ, λ, µ∈Cn.
(a)(κ⊗λ)⊗ µ=κ⊗(λ⊗ µ)
(b) κ⊗λ= λ⊗κ
(c) κ⊗0=0, κ⊗ 1=κ, κ⊗2=κ⊕κ.
(d) κ⊗(λ⊕ µ)=(κ⊗ λ)⊕(κ⊗ µ)
(e) λ≤ µ implies κ⊗λ≤κ⊗ µ.
Proof. (a)�ere is a canonical bijection(A×B)×C→A×(B×C)with
⟨⟨a,b⟩,c⟩↦⟨a,⟨b,c⟩⟩.
(b)�ere is a canonical bijection A×B→B×Awith⟨a,b⟩↦⟨b,a⟩.
(c)A×∅=∅.�ere are canonical bijections
A×{0}→A and A⊍A=[2]×A→ A×[2].
(d)�ere exists a bijection A×(B⊍C)→(A×B)⊍(A×C)with
⟨a,⟨0,b⟩⟩↦⟨0,⟨a,b⟩⟩ and ⟨a,⟨1,c⟩⟩↦⟨1,⟨a,c⟩⟩.
(e) Given an injective function f ∶ B→C we define an injective
function A×B→A×C by⟨a,b⟩↦⟨a, f(b)⟩. ◻
Lemma 3.7. Let κ, λ, µ, ν∈Cn.
118
(a)(κ λ ) µ = κ λ⊗µ
(b)(κ⊗λ) µ = κ µ ⊗ λ µ

(c) κ λ⊕µ = κ λ ⊗ κ µ
(d) κ 0 = 1, κ 1 = κ, κ 2 = κ⊗κ.
(e) If κ≤λand µ≤νthen κ µ ≤ λ ν .
(f) κ

a4. Zermelo-Fraenkelsettheory
β1
9 10 11
4 5
1
0
3
2
8
7
6
β0
15
14
13
12
Figure 1.. Ordering on↓κ×↓κ
Exercise 3.2. Provethatℵ0⊗ℵ0=ℵ0 byshowingthatthe function
is bijective.
↓ω×↓ω→↓ω∶⟨i, k⟩↦ 1 2(i+k)(i+k+ 1)+ k
Westart by computing κ⊗ κ by induction on κ≥ℵ0.
�eorem 3.8. If κ≥ℵ0 then κ⊗ κ=κ.
Proof. We have κ = κ⊗ 1≤κ⊗κ. For the converse, we prove that
κ⊗κ≤ κ by induction on κ.
Notethat,since κ is a cardinal we have α

3. Cardinalarithmetic
is an initial subset of K. If ω≤α

a4. Zermelo-Fraenkelsettheory
4. Cofinality
Frequently, we will construct objects as the union of an increasing sequenceA0⊆
A1⊆ ... ofsets. Inthissectionwewillstudythe cardinality
ofsuchunions.
Definition 4.1. For asequence(κi)i

4. Cofinality
Since∣(↓α) n ∣≤∣↓α∣⊕ℵ0, for n

a4. Zermelo-Fraenkelsettheory
Definition 4.7. (a) Let⟨A,≤⟩ be a linear order. AsubsetX⊆A iscofinal
inA if, for every a∈A,there issome element x∈Xwith a≤x.
We call a function f∶ B→A cofinal ifrng f is cofinal inA.
(b) �ecofinality cf α of an ordinal α is the minimal ordinal β such
thatthere exists a cofinal function f∶↓β→↓α.
Exercise 4.1. Provethat every linear order⟨A,≤⟩ contains a cofinalsubset
X⊆ Asuchthat⟨X,≤⟩ iswell-ordered.
Lemma 4.8. Let⟨A,≤⟩ be a linear order. If X is cofinal in A and Y is
cofinal in X then Y iscofinal in A.
We can restate the definition of the cofinality in a moreuseful form
as follows.
Lemma4.9. If(αi)i

Proof. �e function g∶↓β→↓α with
g(γ)=max{ f(γ), sup{ g(η)+1∣η h(γ)}.
�is function is cofinal since, given η

a4. Zermelo-Fraenkelsettheory
Example. ω andℵ1 areregularwhileℵω issingular.
�e next lemma indicatesthatthe notion of cofinality is mainly interesting
for cardinals.
Lemma4.15. Everyregular ordinal isacardinal.
Proof. Let α ∈ On∖Cn be an ordinal that is not a cardinal and set
κ∶=∣α∣

4. Cofinality
Cardinal exponentiation is the least understood operation of those
introducedso far.�ere are many openquestionsthattheusual axioms
of set theory are not strong enough to answer. For example, we do not
knowwhatthevalue of2 ℵ0 is. Given an arbitrary model ofsettheorywe
can construct a new modelwhere2 ℵ0 =ℵ1, butwe can also find models
where2 ℵ0 equalsℵ2 orℵ3.
Intheremainder of thissection wepresentsome elementary results
thatcan beproved.�e notion of cofinality appears atseveralplaces in
these proofs. First, let us compute the cardinality of all stages Sα, by a
simple induction.
Definition 4.19. We definethe cardinalℶα(κ)(‘bethalpha’), forα∈On
and κ∈ Cn,recursively by
ℶ0(κ)∶= κ ,
ℶα+1(κ)∶=2 ℶα(κ) ,
and ℶδ(κ)∶=sup{ℶα(κ)∣ α κ.
127

a4. Zermelo-Fraenkelsettheory
Proof. Fix a cofinal function f∶↓λ→↓κ. By�eorem 4.6,we have
κ λ =∣(↓κ) ↓λ ∣=∣∏↓κ∣>∣⊍ ↓ f(α)∣≥∣↓κ∣= κ.
α

(c) If λ

a4. Zermelo-Fraenkelsettheory
Hence, κ=ℶδ. Sinceℶδ= κ>ℵ0 we have δ> 0. To show that δ is
a limit suppose that δ=α+1. �enℶα < κ impliesℶδ = 2 ℶα < κ.
Contradiction. ◻
We conclude this section with some results about sets of sequences
indexed by ordinals. Aswewillsee inSection b2.1,such aset formsthe
domain of atree. Recall that asequence indexed by an ordinal α is just
a function↓α→A.
Definition 4.29. IfA is aset and α∈ On,we define
A α ∶= A ↓α
and A

5. �eAxiom ofReplacement
Lemma4.33. If κ isan infiniteregularcardinalthen κ

a4. Zermelo-Fraenkelsettheory
�eorem5.2. �e followingstatements areequivalent:
(1) If F isafunctionandA⊆dom F isasetthen F[A] isalsoaset.
(2) If F isafunctionand dom F isasetthenso is rng F.
(3) A function F isaset if,and only if, dom F isaset.
(4) �ere exists no bijection F∶a→B between a set a and a proper
classB.
(5) AclassA isaset if,and only if,∣A∣

5. �eAxiom ofReplacement
(4)⇒(2) Let F∶A→ B be a function where A= dom F is aset. Let
B0∶= rng F. Since the function F∶a→B0 is surjective there exists
a function G∶B0→a such that F○ G= idB0. Let A0∶= rng G. �e
restriction F∶ A0→ B0 is a bijection. Since A0⊆ A is a set so is B0=
rng F. ◻
Axiom of Replacement. If F isafunction and dom F isaset then so is
rng F.
Letus finallyprovetheresultswepromised intheprecedingsections.
First,upto isomorphism, On isthe onlywell-orderthat is aproper class.
Lemma 5.3. Let A=⟨A,≤A⟩and B=⟨B,≤B⟩bewell-orders. IfAandB
areproperclassesthen A≅B.
Proof. Suppose that A≇B. By �eorem a3.1.11, there either exists an
isomorphism f ∶ A→↓b, for some b∈B, or some isomorphism g∶
↓a→B, for somea∈A. By symmetry,we may assumew.l.o.g.the latter.
↓a is asetsince≤A is le�-narrow. Hence, bythe Axiom ofReplacement,
B= g[↓a] is also aset. Contradiction. ◻
It follows that it does not matter which of the two definitions of an
ordinalwe adopt.
Corollary 5.4. On0≅ Cn≅On.
Finally, we state the general form of the Principle of Transfinite Recursion.
�eorem 5.5 (Principle of Transfinite Recursion). If H∶A

a4. Zermelo-Fraenkelsettheory
Lemma 5.6. Every strictly continuous function f∶ On→On has arbitrarily
large fixed points.
Proof. For every α∈Onwe have to find a fixed point γ≥α. If F is the
fixed-point induction of f over α then F[↓ω] exists. By Lemma a3.3.13
it follows that γ∶= F(∞)= F(ω)≥ α is a fixed point of f . ◻
Corollary 5.7. �erearearbitrarily largecardinals κ suchthat cf κ=ℵ0
andeitherℵκ= κ orℶκ= κ.
Proof. �e functions f∶ α↦ℵα and g∶α↦ℶα arestrictly continuous.
Furthermore, they are defined by transfinite recursion. �erefore, �eorem
5.5 implies that their domain is all of On. By Lemma a3.3.13 and
Lemma 5.6, it follows that f and g have arbitrarily large inductive fixed
points κ, andthese fixedpoints are ofthe form
κ=sup{ f n (α)∣n

6. Conclusion
existence of certain sets. Infinity and Replacement ensure that the cumulative
hierarchy is long enough. �ere are as many stages as there
are ordinals. �e Axioms of Separation and Choice on the other hand
makethe hierarchywide by ensuringthatthepower-set operationyields
enoughsubsets. Inparticular, every definablesubset exists and on every
setthere exists awell-ordering.
Finally, let us note that the usual definition of ZFC is based on a different
axiomatisation wherethe Axiom of Creation isreplaced by four
other axioms and the Axiom of Infinity is stated in a slightly different
way. Nevertheless,we are justified in callingthe abovetheory ZFCsince
thetwovariants are equivalent: every modelsatisfying one ofthe axiom
systems alsosatisfiesthe other one, andviceversa.
135

a4. Zermelo-Fraenkelsettheory
136

Literature
Settheory
M. D.Potter,Sets.An Introduction, OxfordUniversityPress 1990.
A. Lévy,BasicSet�eory,Springer 1979, Dover2002.
K. Kunen,Set�eory. An Introductionto IndependenceProofs, North-Holland
1983.
T. J. Jech,Set�eory, 3rd ed., Springer 2003.
Algebra
P. M. Cohn,UniversalAlgebra,2nd ed.,Springer 1981.
P. M. Cohn,BasicAlgebra,Springer2003.
S. Lang,Algebra, 3rd ed., Springer 2002.
S. MacLane,Categories fortheWorking Mathematician, 2nd ed., Springer 1998.
Topologyand latticetheory
R. Engelking, GeneralTopology, 2nd ed., Heldermann 1989.
C.-A. Faure, A. Frölicher, ModernProjective Geometry, Kluwer2000.
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D.S.Scott,
Continuous LatticesandDomains, CambridgeUniversityPress,2003.
Modeltheory
D. Marker, Model�eory:An Introduction, Springer 2002.
K. Tent and M.Ziegler,ACourse in Model�eory, CambridgeUniversityPress
2012.
logic, algebra & geometry2012-08-08 — ©achim blumensath 1007

Literature
W. Hodges, Model�eory, CambridgeUniversityPress 1993.
B.Poizat,ACourse in Model�eory,Springer2000.
C. C. Chang and H. J. Keisler, Model�eory, 3rd ed., North-Holland 1990.
General modeltheory
J. Barwise and S. Feferman, eds., Model-�eoretic Logics, Springer 1985.
J. T. Baldwin,Categoricity, AMS 2010.
R. Diaconescu, Institution-independent Model�eory, Birkhäuser2008.
H.-D. Ebbinghaus and J. Flum, Finite Model�eory,Springer 1995.
J. Adámek and J.Rosický, LocallyPresentableandAccessibleCategories,
CambridgeUniversityPress 1994.
Stabilitytheory
S. Buechler,EssentialStability�eory, Springer 1996.
E. Casanovas,Simple�eoriesand Hyperimaginaries, CambridgeUniversity
Press2011.
A.Pillay, GeometricStability�eory, OxfordSciencePublications 1996.
F. O.Wagner,Simple�eories, Kluwer AcademicPublishers2000.
S.Shelah,Classification�eory,2nd ed., North-Holland 1990.
1008

SymbolIndex
Chaptera1
S universe ofsets, 5
a∈b membership, 5
a⊆b subset, 5
HF hereditary finitesets,7
⋂A intersection, 11
A∩B intersection, 11
A∖B difference, 11
acc(A) accumulation, 12
fnd(A) foundedpart, 13
⋃A union,21
A∪B union,21
℘(A) powerset,21
cutA cut of A,22
Chaptera2
⟨a0,... ,an−1⟩ tuple,27
A×B cartesianproduct,27
dom f domain of f ,28
rng f range of f ,29
f(a) image of a under f ,29
f∶ A→B function,29
B A set of all functions
f∶ A→ B,29
idA identity function, 30
S○R composition of relations,
30
g○ f composition of functions,
30
R −1 inverse ofR, 30
R −1 (a) inverse image, 30
R∣C restriction, 30
R↾C le� restriction, 31
R[C] image ofC, 31
(ai)i∈I sequence, 37
∏i Ai
pri product, 37
projection, 37
ā sequence, 38
⊍i Ai disjointunion, 38
A⊍B disjointunion, 38
ini insertion map, 39
⇓X initial segment, 41
⇑X final segment, 41
↓X initial segment, 41
↑X final segment, 41
[a,b] closed interval, 41
(a,b) open interval, 41
maxX greatest element, 42
minX minimal element, 42
supX supremum, 42
inf X infimum, 42
logic, algebra & geometry2012-08-08 — ©achim blumensath 1009

Symbol Index
A≅B isomorphism, 44
fix f fixedpoints, 48
lfp f least fixedpoint, 48
gfp f greatest fixedpoint, 48
[a]∼ equivalence class, 54
A/∼ set of∼-classes, 54
TC(R) transitive closure, 55
Chaptera3
a + successor, 59
ord(A) ordertype,64
On class of ordinals,64
On0 von Neumann ordinals,69
ρ(a) rank,73
A

Emb(Σ) category of embeddings,
153
Set∗
category of pointed sets,
153
Set 2 category of pairs, 153
Cat ∗ dual category, 155
F≅G natural isomorphism, 158
Cong(A) set of congruencerelations,
161
Cong(A) congruence lattice, 161
A/∼ quotient, 165
Chapterb2
∣x∣ length of asequence, 173
x⋅y concatenation, 173
⪯ prefix order, 173
≤lex lexicographic order, 173
∣v∣ level of avertex, 176
frk(v) foundationrank, 178
a⊓b infimum, 181
a⊔b supremum, 181
a ∗ complement, 184
L op dual lattice, 190
cl↓(X) ideal generated by X, 190
cl↑(X) filter generated by X, 190
B2
two-element boolean
algebra, 195
ht(a) height of a,200
rkP(a) partitionrank,204
deg P (a) partition degree,208
Chapterb3
Symbol Index
T[Σ,X] finiteΣ-terms,213
tv subterm atv,214
free(t) freevariables,217
t A [β] value of t,217
T[Σ,X] term algebra,218
t[x/s] substitution,220
SigVar category ofsignatures and
variables,221
Sig category ofsignatures,222
Var category ofvariables,222
Term category ofterms,222
A∣µ µ-reduct of A, 223
Str[Σ] class ofΣ-structures, 223
Str[Σ,X] class of allΣ-structures
withvariable
assignments, 223
StrVar category of structures and
assignments, 223
Str category of structures, 223
∏i A i direct product,225
⟦φ⟧ set of indices,227
ā∼u ¯ b filter equivalence,227
u∣J restriction of uto J,228
∏i A i /u reducedproduct,228
A u ultrapower,229
lim
�→ D direct limit, 237
⋃i A i union of a chain,241
lim
←� D inverse limit,241
Chapterb4
cl(A) closure of A,245
int(A) interior of A,245
1011

Symbol Index
∂A boundary of A,245
rkCB(x/A) Cantor-Bendixsonrank,
267
spec(L) spectrum of L,272
⟨x⟩ basic closedset,272
clop(S) algebra of clopensubsets,
276
Chapterb5
Aut M automorphism group,288
G/U set of cosets,288
G/N factor group,290
Sym Ω symmetric group,291
ga action of g on a,292
Gā orbit of ā,292
G(X) pointwisestabiliser,293
setwise stabiliser, 293
G{X}
dclG(U) G-definitional closure,297
aclG(U) G-algebraic closure,297
⟨ā↦ ¯ b⟩ basic openset ofthe group
topology,298
degp degree, 302
Idl(R) lattice of ideals, 303
R/a quotient of aring, 305
Ker h kernel, 305
spec(R) spectrum, 306
⊕i Mi directsum, 309
M (I) direct power, 309
dim V dimension, 312
FF(R) field of fractions, 314
K(ā) subfield generated by ā, 317
p[x] polynomial function, 319
Aut(L/K)automorphisms over K,
326
1012
∣a∣ absolutevalue, 329
Chapterc1
ZL[K,X] Zariski logic, 345
⊧ satisfaction relation, 346
BL(B) boolean logic, 346
FOκℵ0[Σ,X] infinitary first-order
logic, 347
¬φ negation, 347
⋀Φ conjunction, 347
⋁Φ disjunction, 347
∃xφ existential quantifier, 347
∀xφ universalquantifier, 347
FO[Σ,X] first-order logic, 347
A⊧φ[β] satisfaction, 348
true true, 349
false false, 349
φ∨ψ disjunction, 349
φ∧ψ conjunction, 349
φ→ψ implication, 349
φ↔ψ equivalence, 349
free(φ) freevariables, 352
qr(φ) quantifierrank, 355
ModL(Φ) class of models, 356
Φ⊧φ entailment, 362
≡ logical equivalence, 362
Φ ⊧ closureunder entailment,
362
�L(J) L-theory, 363
≡L L-equivalence, 364
dnf(φ) disjunctive normal form,
369
cnf(φ) conjunctive normal form,
369

nnf(φ) negation normal form, 371
Logi$ category of logics, 380
∃ λ xφ cardinality quantifier, 383
FOκℵ0(wo) FOwithwell-ordering
quantifier, 383
W well-ordering quantifier,
383
QK Lindström quantifier, 384
SOκℵ0[Σ,Ξ] second-order logic, 385
MSOκℵ0[Σ,Ξ] monadic
second-order logic, 385
PO category of partial orders,
390
Lb Lindenbaum functor, 390
¬φ negation, 391
φ∨ψ disjunction, 391
φ∧ψ conjunction, 392
L∣Φ restriction toΦ, 393
L/Φ localisation toΦ, 393
⊧Φ consequence moduloΦ,
393
≡Φ equivalence moduloΦ, 393
Chapterc2
EmbL(Σ) category of L-embeddings,
395
QFκℵ0 [Σ,X] quantifier-free
formulae, 396
∃∆ existential closure of∆, 396
∀∆ universal closure of ∆, 396
∃κℵ0 existential formulae, 396
∀κℵ0 universal formulae, 396
∃ + κℵ0 positive existential
formulae, 396
Symbol Index
⪯∆ ∆-extension, 400
⪯ elementary extension, 400
Φ ⊧ ∆ ∆-consequences ofΦ, 422
≤∆ preservation of
∆-formulae, 422
Chapterc3
S(L) set oftypes, 427
⟨Φ⟩ types containingΦ, 427
tpL (ā/M)L-type of ā, 428
S ¯s L(T) typespace for atheory, 428
S ¯s L(U) typespace overU, 428
S(L) typespace, 433
f(p) conjugate of p, 443
S∆(L) S(L∣∆)with topology
induced from S(L), 456
⟨Φ⟩∆ closedset in S∆(L), 456
p∣∆ restrictionto∆, 459
tp∆ (ā/U)∆-type of ā, 459
Chapterc4
≡α α-equivalence, 473
≡∞ ∞-equivalence, 473
pIso κ (A, B) partial isomorphisms,
474
ā↦ ¯ b map ai↦ bi, 474
∅ the empty function, 474
Iα(A, B) back-and-forthsystem, 475
I∞(A, B) limit ofthesystem, 477
≅α α-isomorphic, 477
≅∞ ∞-isomorphic, 477
m=k n equalityupto k, 480
1013

Symbol Index
φ α A,ā
Hintikka formula, 482
EFα(A,ā, B, ¯ b)
Ehrenfeucht-Fraïssé
game, 485
EF κ ∞(A,ā, B, ¯ b)
Ehrenfeucht-Fraïssé
game, 485
I κ FO(A, B)partial FO-maps ofsize κ,
494
⊑ κ iso ∞κ-simulation, 495
≅ κ iso ∞κ-isomorphic, 495
A⊑ κ 0 B I κ 0(A, B)∶A⊑ κ iso B, 495
A≡ κ 0 B I κ 0(A, B)∶A≡ κ iso B, 495
A⊑ κ FO B I κ FO(A, B)∶A⊑ κ iso B, 495
A≡ κ FO B I κ FO(A, B)∶A≡ κ iso B, 495
A⊑ κ ∞ B I κ ∞(A, B)∶ A⊑ κ iso B, 495
A≡ κ ∞ B I κ ∞(A, B)∶ A≡ κ iso B, 495
G(A) Gaifman graph, 501
Chapterc5
L≤ L ′ L ′ is as expressive as L, 509
(a) algebraic, 510
(b) boolean closed, 510
(b+) positive boolean closed,
510
(c) compactness, 510
(cc) countable compactness,
510
(fop) finite occurrenceproperty,
510
(kp) Karpproperty, 510
(lsp) Löwenheim-Skolem
property, 510
(rel) closedunder
1014
relativisations, 510
(sub) closedundersubstitutions,
510
(tup) Tarskiunionproperty, 510
hnκ(L) Hanf number, 514
lnκ(L) Löwenheim number, 514
wnκ(L) well-ordering number, 514
occ(L) occurrence number, 514
prΓ (K) Γ-projection, 532
PCκ(L,Σ) projective L-classes, 533
L0≤ κ pc L1 projective reduction, 533
RPCκ(L,Σ) relativised projective
L-classes, 537
L0≤ κ rpc L1 relativised projective
reduction, 537
∆(L) interpolation closure, 545
ifp f inductive fixedpoint, 554
lim inf f leastpartial fixedpoint, 554
limsup f greatestpartial fixedpoint,
554
fφ function defined byφ, 561
FOκℵ0(LFP) least fixed-point logic,
561
FOκℵ0(IFP) inflationary fixed-point
logic, 562
FOκℵ0(PFP) partial fixed-point
logic, 562
⊲φ stage comparison, 572
Chapterd1
tor(G) torsionsubgroup,603
a/n divisor, 604
DAG theory of divisible
torsion-free abelian

groups,604
ODAG theory of ordered divisible
abelian groups,604
div(G) divisible closure,605
F field axioms,608
ACF theory of algebraically
closed fields,608
RCF theory ofreal closed fields,
609
Chapterd2
(

Symbol Index
Chapter e3
Subκ(A) substructures of A,759
atp(ā) atomictype,765
ηpq extension axiom,765
T[K] extension axioms forK,
766
Tran[Σ] randomtheory,766
κn(φ) number of models,768
Pr n
M[M⊧φ] density of models,768
Chapter e4
increasing κ-tuples,773
κ→(µ) ν
λ partition theorem, 773
pf(η,ζ) prefix ofζ of length∣η∣, 778
T∗(κ

(mon) Monotonicity, 928
(nor) Normality, 928
(lrf) Le�Reflexivity, 928
(ltr) Le�Transitivity, 928
(fin) Finite Character, 929
(sym) Symmetry, 929
(rbm) Right Base Monotonicity,
929
(srb) StrongRight Boundedness,
929
cl √ closure operation
associatedwith √ , 934
(inv) Invariance, 941
(sfin) Strong Finite Character,
941
(ext) Extension, 941
(lbd) Le� Boundedness, 941
(rbd) Right Boundedness, 941
lb( √ ) le� boundedness cardinal
of √ , 941
rb( √ ) right boundedness
cardinal of √ , 941
A df√ U B definable over, 943
Symbol Index
A at√ U B isolated over, 943
A s√
U B non-splitting over, 943
A u√
U B finitely satisfiable, 948
Av(u/B) average type of u, 948
√
∗
forking relation to √ , 953
A i√
U B globally invariant over, 956
Chapter f3
A d√
U B non-dividing, 965
A f√
U B non-forking, 965
(ind) Independence �eorem,
992
ā∼ ls U ¯ b indiscernible sequence
startingwith ā, ¯ b,... ,
993
ā≡ ls U ¯ b Lascar strong type
equivalence, 993
1017

Symbol Index
1018

Index
abelian group,287
abstract elementary class, 839
abstract independence relation, 928
accumulation, 12
accumulationpoint,266
action,292
acyclic, 420
addition of cardinals, 116
addition of ordinals, 89
adjoint functors,220
affine geometry, 881
aleph, 115
algebraic, 139, 712
Aut-algebraic,712
G-algebraic,297
algebraic class, 840
algebraic closure,297,712
algebraic closure operator, 51
algebraic diagram, 401
algebraic elements, 322
algebraic field extensions, 322
algebraic logic, 389
algebraic prime model, 592
algebraically closed, 712
algebraically closed field, 322, 608
algebraically independent, 321
almost strongly minimal theory, 900
amalgamation class, 849
amalgamation property, 760, 848
amalgamation square, 549
Amalgamation�eorem, 422
antisymmetric, 40
arity,28,29, 139
associative, 31
asynchronousproduct,651
atom, 347
atom of a lattice, 200
atomic, 200
atomic diagram, 401
atomic structure,721
atomictype,765
atomless,200
automorphism, 146
automorphism group,288
averagetype,791
averagetype of an
Ehrenfeucht-Mostowski
functor, 830
average type of an indiscernible
system, 797
average type of anultrafilter, 948
Axiom of Choice, 109, 360
Axiom of Creation, 19, 360
Axiom of Extensionality, 5, 360
logic, algebra & geometry2012-08-08 — ©achim blumensath 1019

Index
Axiom of Infinity, 24, 360
Axiom of Replacement, 133, 360
Axiom of Separation, 10, 360
axiom system, 356
axiomatisable, 356
axiomatise, 356
back-and-forth property, 474
back-and-forth system, 474
Baire, property of —, 265
ball, 244
base, closed —, 246
base, open —, 246
basic Horn formula,634
basis, 110, 878, 881
beth, 127
Beth property, 544, 717
bidefinable, 751
biinterpretable, 757
bijective, 31
boolean algebra, 184, 357, 392
boolean closed, 392
boolean lattice, 184
boolean logic, 346, 364
boundvariable, 352
boundary,245,656
κ-bounded, 494
bounded independencerelation, 941
bounded lattice, 181
bounded linear order, 479
bounded logic, 514
boundedness, 941
box,656
branch, 175
branching degree, 177
Cantor discontinuum,253, 434
1020
Cantor normal form, 100
Cantor-Bendixson rank, 267, 279
cardinal, 113
cardinal addition, 116
cardinal exponentiation, 116, 127
cardinal multiplication, 116
cardinality, 113
cardinality quantifier, 384
cartesian product,27
categorical,743
category, 152
¯δ-cell, 671
cell decomposition, 673
Cell Decomposition �eorem, 674
chain, 42
L-chain, 403
chain of structures,241
chaintopology,252
chain-bounded formula, 993
Chang’s reduction, 432
character, 105
characteristic, 609
characteristic of a field, 316
choice function, 106
Choice, Axiom of —, 109, 360
class, 9, 54
clopen set, 243
=-closed, 413
closed base, 246
closed function,248
closed interval,655
closedset, 51, 53, 243
closedsubbase,246
closedunbounded, 558
closedunderrelativisations, 510
closedundersubstitutions, 510
closure operator, 51, 110

closure ordinal, 81
closurespace, 53
closureunderreverseultrapowers,
633
closure,topological —,245
cocone,242
codirected, 232
coefficient, 302
cofinal, 124
cofinality, 124
Coincidence Lemma, 217
commutative,287
commutativering, 300
commuting diagram, 154
comorphism of logics, 380
compact, 254, 509
compact, countably —, 509
Compactness�eorem, 416, 431
compactness theorem, 618
compatible, 375
complement, 184
complete, 364
κ-complete, 494
complete partial order, 43, 50, 53
complete type, 427
composition, 30
concatenation, 173
condition of filters, 621
cone, 238
congruencerelation, 161
conjugacy class,293
conjugation,293
conjunction, 347, 391
conjunctive normal form, 369
connected, definably —, 659
consequence, 362, 390, 422
Index
consistence of filterswith conditions,
621
consistent, 356
constant, 29, 139
constructedset,735
construction,735
continuous, 46, 134, 248
contradictory formulae, 524
contravariant, 157
coset,288
countable, 110, 115
countably compact, 509
covariant, 157
cover,254
Creation, Axiom of —, 19, 360
cumulative hierarchy, 18
cut,22
deciding a condition,621
definable,712
Aut-definable,712
G-definable,297
definable expansion, 375
definable structure,751
definabletype, 466, 943
definablewith parameters, 657
definably connected, 659
defining a set, 349
definition of a type, 465
definitional closed, 712
definitional closure,297,712
degree of apolynomial, 302
dense class, 1001
dense linear order, 496
κ-dense linear order, 496
dense order, 357
dense set, 263
1021

Index
dense sets in directed orders, 233
dependence relation, 875
dependent, 875
dependent set, 110
derivation, 301
diagram, 236, 241
L-diagram, 401
Diagram Lemma, 401, 530
difference, 11
dimension, 881
dimension function, 882
dimension of a cell,671
dimension of avectorspace, 312
direct limit, 237
direct power, 309
direct product,225
directsum of modules, 309
directed, 232
directed diagram, 236
discontinuum,253
discrete linear order, 479
discrete topology, 244
disintegrated matroid, 888
disjointunion, 38
disjunction, 347, 391
disjunctive normal form, 369
distributive, 184
dividing, 965
dividing chain, 977
dividing κ-tree, 985
divisible closure,605
divisible group,604
domain,28, 141
dual category, 155
dual lattice, 190
Ehrenfeucht-Fraïssé game, 485, 488
1022
Ehrenfeucht-Mostowski functor, 830,
846
Ehrenfeucht-Mostowski model, 830
element of a set, 5
elementary diagram, 401
elementary embedding, 395, 400
elementary extension, 400
elementary map, 395
elementarysubstructure, 400
eliminationset, 588
embedding, 44, 146, 237, 396
∆-embedding, 395
K-embedding, 839
elementary —, 395
embedding of a tree into a lattice, 206
embedding of logics, 380
embedding of permutation groups,
752
embedding, elementary —, 400
endomorphismring, 307
entailment, 362, 390
epimorphism, 155
equivalence class, 54
equivalence of categories, 158
equivalencerelation, 54, 357
L-equivalent, 364
α-equivalent, 473, 488
equivalent formulae, 362
Erdős-Rado theorem, 776
Euklidean norm,243
even, 770
exchange property, 110
existential, 396
existential closure, 597
existentialquantifier, 347
existentially closed, 597
expansion, 145, 842

expansion, definable —, 375
explicit definition, 544
exponentiation of cardinals, 116, 127
exponentiation of ordinals, 89
extension, 142, 941
∆-extension, 400
extension axiom, 765
extension of fields, 317
extension, elementary —, 400
Extensionality, Axiom of —, 5, 360
factorisation, 165
Factorisation Lemma, 148
family, 37
field, 300, 359, 400, 608
field extension, 317
field of a relation, 29
field of fractions, 314
field, real —, 329
field, real closed —, 332
filter, 189, 193, 430
final segment, 41
κ-finitary set of partial isomorphisms,
494
finite, 115
finite character, 51, 105, 929
finite character, strong —, 941
finite intersection property, 197
finite occurrenceproperty, 509
finite, being — over aset,674
finitely axiomatisable, 356
finitely branching, 177
finitely generated, 144
finitely satisfiable type, 948
first-order interpretation, 348, 377
first-order logic, 347
fixed point, 48, 81, 134, 553
fixed-point induction,77
fixed-pointrank, 572
follow, 362
forcing, 621
forgetful functor, 157,220
forking chain, 977
forkingrelation, 941
formalpowerseries, 301
formula, 346
foundationrank, 178
founded, 13
Fraïssé limit, 764
free algebra, 218
free extension of types, 942
√
-free extension of types, 942
free model, 639
free structures,648
freevariables,217, 352
function,29
functional,29, 139
functor, 157
Index
Gaifman graph, 501
Gaifman,�eorem of —, 507
Galoissaturatedstructure, 855
Galoisstable, 855
Galoistype, 841
game,79
generalisedproduct,650
κ-generated, 239, 759, 809
generatedsubstructure, 144
generated, finitely —, 144
generating, 41
generating an ideal, 304
generator, 144, 639
geometric dimension function, 882
geometric independencerelation, 929
1023

Index
geometry, 881
graduatedtheory, 596,681
graph, 39
greatest element, 41
greatest fixed point, 553
greatest lower bound, 42
greatestpartial fixedpoint, 554
group, 34, 287, 358
group action,292
group, ordered —,604
guard, 349
Hanf number, 514, 534, 847
Hanf’s �eorem, 502
Hausdorffspace,253
height, 176
height in a lattice, 200
Henkin property, 724
Henkin set, 724
Herbrand model, 412, 724
hereditary, 12
κ-hereditary, 760, 809
hereditary finite, 7
Hintikka formula, 482, 483
Hintikka set, 414, 724, 725
history, 15
homeomorphism, 248
homogeneous,685,773
≈-homogeneous,779
κ-homogeneous, 500,685
homogeneous matroid, 888
homomorphic image, 147,643
homomorphism, 146, 396
Homomorphism �eorem, 168
homotopic interpretations, 756
Horn formula,634
1024
ideal, 189, 193, 303
identity, 153
image, 31
implication, 349
implicit definition, 544
inclusion morphism, 393
inconsistent, 356
k-inconsistent, 965
increasing, 44
independence property, 799
independence relation, 928
independence relation of a matroid,
927
Independence �eorem, 992
independent, 875
independent set, 110, 881
index of asubgroup,288
indiscerniblesequence,789
indiscerniblesystem,796
inducedsubstructure, 142
inductive,77
inductive fixedpoint, 81, 553, 554
inductively ordered, 81, 105
infimum, 42, 181
infinitary first-order logic, 347
infinitary second-order logic, 385
infinite, 115
Infinity, Axiom of —, 24, 360
inflationary, 81
inflationary fixed-point logic, 562
initial object, 156
initial segment, 41
injective, 31
κ-injective structure, 852
innervertex, 175
insertion, 39
inspired by, 797

integral domain, 314, 611
interior, 245, 656
interpolant, 549
interpolation closure, 545
interpolationproperty, 543
∆-interpolation property, 543
interpretation, 346, 348, 377
intersection, 11
interval, 655
invariance, 941
invariant class, 1001
invariant type, 956
inverse, 30, 155
inverse diagram, 241
inverse limit, 241
inverse reduct, 819
irreduciblepolynomial, 320
irreflexive, 40
isolated point, 266
isolated type, 721, 943
isomorphic, 44
α-isomorphic, 477, 488
isomorphic copy, 643
isomorphism, 44, 146, 155, 158, 396
isomorphism, partial —, 473
joint embedding property, 760, 849
Jónsson class, 849
Karp property, 509
kernel, 148
kernel of a ring homomorphism, 305
label, 213
largesubsets,720
Lascarstrongtype, 993
lattice, 181, 357, 392
leaf, 175
least element, 41
least fixed point, 553
least fixed-point logic, 561
least partial fixed point, 554
leastupper bound, 42
le� bounded, 941
le� ideal, 303
le� reflexivity, 928
le� restriction, 31
le� transitivity, 928
le�-narrow, 57
length, 173
level, 176
level embedding function,779
levels of atuple,779
lexicographic order, 173, 869
li�ing functions, 551
limit, 59
limitstage, 19
Lindenbaum algebra, 390
Lindenbaum functor, 390
Lindström quantifier, 384
linear independence, 309
linear matroid, 881
linear order, 40
linear representation, 585
literal, 347
local, 504
local enumeration,670
κ-local functor, 809
localisation morphism, 393
localisation of a logic, 393
locally compact, 254
locally finite matroid, 888
locally modular matroid, 888
logic, 346
Index
1025

Index
logical system, 387
Łoś’ theorem, 615
Łoś-Tarski �eorem, 584
Löwenheim number, 514, 534, 537, 839
Löwenheim-Skolem property, 509
Löwenheim-Skolem-Tarski �eorem,
421
lower bound, 42
lower fixed-point induction, 554
map,29
∆-map, 395
map, elementary —, 395
mapping, 29
matroid, 880
maximal element, 41
maximal ideal, 314
maximal ideal/filter, 189
meagre, 264
membership relation, 5
minimal, 13, 57
minimal element, 41
minimal polynomial, 323
minimal rank and degree, 208
minimal set, 893
model, 346
model companion, 597
model of a presentation, 639
model-complete, 597
κ-model-homogeneousstructure, 852
modular, 184
modular lattice,200
modular law,202,203
modular matroid, 888
modularity, 938
module, 306
monadic second-order logic, 385
1026
monoid, 31, 175, 287
monomorphism, 155
monotone, 656
monotonicity, 928
monster model, 720
Morley degree, 920
Morley rank, 917
Morley sequence, 958
Morley-free extension of atype, 920
morphism, 152
morphism of logics, 380
morphism of matroids, 888
morphism of permutation groups,751
multiplication of cardinals, 116
multiplication of ordinals, 89
natural isomorphism, 158
naturaltransformation, 158
negation, 347, 391
negation normal form, 371
negative occurrence, 561
neighbourhood,243
neutral element, 31
node, 175
normalsubgroup,289
normality, 928
nowhere dense,264
o-minimal,658, 802
object, 152
occurrence number, 514
oligomorphic,292,743
omitting a type, 428
omitting types, 432
open base, 246
open cover, 254
open dense order, 357

open interval, 655
open set, 243
opensubbase,247
orbit,292
order, 356
order property, 463
order topology, 251, 656
order type, 64, 789
orderable ring, 329
ordered group,604
orderedpair,27
orderedring, 329
ordinal, 64
ordinal addition, 89
ordinal exponentiation, 89
ordinal multiplication, 89
ordinal,von Neumann —,69
package,737
pair, 27
parameter-definable, 657
partial fixed point, 554
partial fixed-point logic, 562
partial function,29
partial isomorphism, 473
partial isomorphism modulo a filter,
626
partial order, 40, 356
partial order, strict —, 40
partition, 55, 204
partition degree, 208
partition rank, 204
partitioning a relation, 674
path, 175
Peano Axioms, 386
pinning down, 514
point, 243
Index
polynomial, 302
polynomial function, 319
polynomial ring, 302
positive existential, 396
positive occurrence, 561
positiveprimitive,634
power set, 21
predicate, 28
predicate logic, 346
prefix, 173
prefix order, 173
preforking relation, 941
prelattice, 193
prenex normal form, 371
preorder, 192, 390
presentation, 639
preservation by a function, 395
preservation in products,634
preservation insubstructures, 398
preservation inunions of chains, 399
preserving fixed points, 551
prime field, 316
prime ideal, 194, 306
prime model, 734
prime model, algebraic, 592
primitive formula, 597
principal ideal/filter, 189
Principle ofTransfiniteRecursion,75,
133
product,27, 37, 643
product of categories, 156
product of linear orders, 86
producttopology,259
product, direct —,225
product, generalised —,650
product,reduced —,228
product,subdirect —,226
1027

Index
projection, 37, 532
projective class, 532
projective geometry, 887
projectively reducible, 533
projectively κ-saturated,702
proper, 189
property of Baire,265
pseudo-elementary, 533
pseudo-saturated,705
quantifier elimination, 588,610
quantifierrank, 355
quantifier-free, 355
quantifier-free formula, 396
quasivariety,643
quotient, 165
Rado graph,766
Ramsey’stheorem,774
random graph,766
randomtheory,766
range,29
rank,73, 178
∆-rank, 917
rank, foundation –, 178
real closed field, 332, 609
real closure of a field, 332
real field, 329
realising a type, 428
reducedproduct,228,643
reduct, 145
µ-reduct, 223
reflexive, 40
regular, 125
regular filter,617
regular logic, 510
relation,28
1028
relational, 139
relationalvariant of a structure, 821
relativisation, 376, 510
relativised projective class, 537
relativised projectively reducible, 537
relativised quantifiers, 349
relativised reduct, 537
Replacement, Axiom of —, 133, 360
replica functor, 823
restriction, 30
restriction of a filter, 228
restriction of a Galois type, 859
restriction of a logic, 393
restriction of a type, 459
retract of a logic, 446
retraction of logics, 446
reverseultrapower,633
right base monotonicity, 929
right bounded, 941
ring, 300, 359
ring, orderable —, 329
ring, ordered —, 329
root, 175
root of a polynomial, 319
Ryll-Nardzewski �eorem, 743
satisfaction, 346
satisfaction relation, 346, 348
satisfiable, 356
saturated,691
κ-saturated, 565,691
κ-saturated,projectively —,702
Scott height, 483
Scott sentence, 483
second-order logic, 385
segment, 41
semantics functor, 387

semantics of first-order logic, 348
semi-strict homomorphism, 146
semilattice, 181
sentence, 352
separated formulae, 524
Separation, Axiom of —, 10, 360
sequence, 37
signature, 139, 141, 221, 222
simple structure, 315
simple theory, 975
simply closed, 592
singular, 125
skew embedding, 786
skew field, 300
Skolem axiom, 406
Skolem expansion, 843
Skolem function, 406
Skolemtheory, 406
Skolemisation, 406
smallsubsets,720
sort, 141
spanning, 878
special model,705
specification of a dividing chain, 977
specification of a dividing κ-tree, 985
specification of a forking chain, 977
spectrum,272, 431, 434
spectrum of aring, 306
spine, 825
stabiliser, 293
κ-stable, 463
stage, 15, 77
stage comparison relation, 572
Stone space, 276, 431, 434
strict homomorphism, 146
strict Horn formula,634
strict∆-map, 395
strict order property, 804
strict partial order, 40
strictly increasing, 44
strictly monotone, 656
strong γ-chain, 861
strong γ-limit, 861
strong finite character, 941
strong limit cardinal, 705
strong right boundedness, 929
strongly homogeneous,685
strongly κ-homogeneous,685
strongly local functor, 825
strongly minimalset, 893
strongly minimal theory, 900, 990
structure, 139, 141, 223
subbase, closed —,246
subbase, open —,247
subcategory, 156
subcover,254
subdirectproduct,226
subdirectly irreducible,226
subfield, 316
subformula, 352
subset, 5
subspacetopology,248
subspace, closure —,248
substitution,220, 367, 510
substructure, 142,643, 759, 809
∆-substructure, 400
K-substructure, 840
substructure, elementary —, 400
substructure, generated —, 144
substructure, induced —, 142
subterm,214
subtree, 176
successor, 59, 175
successorstage, 19
Index
1029

Index
sum of linear orders, 85
superset, 5
supremum, 42, 181
surjective, 31
symbol, 139
symmetric, 40
symmetric group,291
symmetric independencerelation,
929
syntax functor, 387
Tarskiunionproperty, 510
tautology, 356
term, 213
term algebra, 218
term domain, 213
term,value of a —,217
term-reduced, 368
terminal object, 156
L-theory, 363
theory of a functor, 815
topological closure,245,656
topological closure operator, 51,245
topological group,298
topologicalspace,243
topology, 243
topology of the type space, 433
torsion element, 603
torsion-free, 603
total order, 40
totally disconnected, 253
totally indiscernible sequence,790
transcendence basis, 322
transcendence degree, 322
transcendental elements, 322
transcendental field extensions, 322
transfinite recursion,75, 133
1030
transitive, 12, 40
transitive action, 292
transitive closure, 55
transitive dependencerelation, 875
transitivity, le� —, 928
tree, 175
treeproperty, 984
tree-indiscernible,797
trivial filter, 189
trivial ideal, 189
trivialtopology,244
tuple,28
Tychonoff,�eorem of —,261
type, 459
L-type, 427
Ξ-type,702
α-type, 428
¯s-type, 428
type of a function, 141
type of arelation, 141
typespace, 433
type topology, 433
type, average —, 791
type, average — of an indiscernible
system, 797
type, complete —, 427
type, Lascar strong —, 993
types of dense linear orders, 429
ultrafilter, 194, 430
ultrahomogeneous,761
ultrapower,229
ultraproduct,229,695
unbounded class, 847
uncountable, 115
uniform dividing chain, 978
uniform dividing κ-tree, 985

uniform forking chain, 978
uniformly finite, being — over aset,
674
union,21
union of a chain,241, 403, 586
unit of aring, 314
universal, 396
κ-universal, 691,760
universalquantifier, 347
universalstructure, 852
universe, 139, 141
unsatisfiable, 356
unstable, 463
upper bound, 42
upper fixed-point induction, 554
valid, 356
value of aterm,217
variable,222
variablesymbols, 347
variables, free —, 217, 352
variety, 643
Vaughtianpair, 901
vectorspace, 306
vertex, 175
von Neumann ordinal,69
Index
weak γ-chain, 861
weak γ-limit, 861
weak homomorphic image, 147,643
weakly regular logic, 510
well-founded, 13, 57, 81, 109
well-order, 57, 109, 133, 494
well-ordering number, 514, 534
well-ordering quantifier, 383, 384
winning strategy, 486
word construction, 816, 821
Zariski logic, 345
Zariski topology, 244
zero-dimensional, 253
zero-divisor, 314
Zero-One Law, 770
ZFC, 359
Zorn’s Lemma, 110
1031

1032
�e Roman and Fraktur alphabets
A a A a N n N n
B b B b O o O o
C c C $ P p P p
D d D d Q q Q q
E e E e R r R r
F f F f S s S s +
G g G g T t T t
H h H h U u U u
I i J i V v V v
J j J j W w W w
K k K k X x X x
L l L l Y y Y y
M m M m Z z Z z
�e Greek alphabet
A α alpha N ν nu
B β beta Ξ ξ xi
Γ γ gamma O o omicron
∆ δ delta Π π pi
E ε epsilon P ρ rho
Z ζ zeta Σ σ sigma
H η eta T τ tau
Θ ϑ theta Υ υ upsilon
I ι iota Φ ϕ phi
K κ kappa X χ chi
Λ λ lambda Ψ ψ psi
M µ mu Ω ω omega