Table Of ContentRUSSIANACADEMYOFSCIENCES MINISTRYOFEDUCATIONANDSCIENCE
OFTHERUSSIANFEDERATION
VLADIKAVKAZSCIENTIFICCENTER
SOUTHERNMATHEMATICALINSTITUTE NORTHOSSETIANSTATEUNIVERSITY
TRENDS IN SCIENCE • THE SOUTH OF RUSSIA
A MATHEMATICAL MONOGRAPH
Issue 6
BOOLEAN VALUED ANALYSIS:
SELECTED TOPICS
by
A. G. Kusraev and S. S. Kutateladze
Vladikavkaz
2014
ББК 22.16
УДК517.98
K94
Editor
A.E. Gutman
Reviewers:
S.A. Malyugin and E.Yu. Emelyanov
Editors of theseries:
Yu.F. Korobeinik and A.G. Kusraev
Kusraev A. G. and Kutateladze S. S.
Boolean Valued Analysis: Selected Topics / Ed. A.E. Gutman.—Vladi-
kavkaz: SMI VSC RAS, 2014.—iv+400 p.—(Trends in Science: The South of
Russia. A Mathematical Monograph. Issue6).
The book treats Boolean valued analysis. This term signifies the technique of
studying properties of an arbitrary mathematical object by means of comparison
between its representations intwo different set-theoretic models whose construction
utilizes principally distinct Boolean algebras. As these models, we usually take the
classicalCantorianparadiseintheshapeofthevonNeumannuniverseandaspecially-
trimmedBooleanvalueduniverseinwhichtheconventionalset-theoreticconceptsand
propositions acquire bizarre interpretations. Exposition focuses on the fundamental
properties oforderboundedoperators invectorlattices.Thisvolumeisintendedfor
theclassicalanalystseekingnewpowerful toolsandforthemodeltheoristinsearch
ofchallengingapplicationsofnonstandard modelsofsettheory.
Кусраев А. Г., Кутателадзе С. С.
Булевозначный анализ: Избранные темы / отв. ред. А. Е. Гутман.—
Владикавказ:ЮМИ ВНЦ РАНиРСО-А,2014.—iv+400 с.—(Итоги науки.
Юг России. Математическая монография. Вып. 6).
Монографияпосвященабулевозначномуанализу.Такназываютаппаратис-
следованияпроизвольныхматематическихобъектов,основанныйнасравнитель-
ном изучении их вида в двух моделях теории множеств, конструкции которых
основаны на принципиально различных булевых алгебрах. В качестве этих мо-
делейфигурируютклассическийкантороврайвформеуниверсумафонНейма-
на и специально построенный булевозначный универсум, в котором теоретико-
множественные понятия и утверждения получают весьма нетрадиционные тол-
кования. Основное внимание уделено фундаментальным свойствам порядково
ограниченных операторов в векторных решетках. Книга ориентирована на ши-
рокий круг читателей, интересующихся современными теоретико-модельными
методамивихприложениикфункциональному анализу.
ISBN 978-5-904695-24-8 (cid:2)c Southern MathematicalInstitute
VSCRAS&RNO-A,2014
(cid:2)c A.G.Kusraev,2014
(cid:2)c S.S.Kutateladze, 2014
PREFACE
Humans definitely feel truth but cannot define truth properly. That
iswhatAlfredTarskiexplainedtousinthe 1930s.Mathematicspursues
truthbywayofproof,aswittilyphrasedbySaundersMacLane.Boolean
valued analysis is one of the vehicles of the pursuit, resulting from the
fusion of analysis and model theory.
Analysis is the technique of differentiation and integration. Diffe-
rentiation discovers trends, and integration forecasts the future from
trends. Analysis opens ways to understanding of the universe.
Model theory evaluates and counts truth and proof. The chase of
truth not only leads us close to the truth we pursue but also enables us
to nearly catch up with many other instances of truth which we were
not aware nor even foresaw at the start of the rally pursuit. That is
what we have learned from Boolean valued models of set theory. These
models stem from the famous works by Paul Cohen on the continuum
hypothesis. They belong to logic and yield a profusion of the surprising
and unforeseen visualizations of the ingredients of mathematics. Many
promising opportunities are open to modeling the powerful habits of
reasoning and verification.
Logicorganizesandordersourwaysofthinking,manumittingusfrom
conservatism in choosing the objects and methods of research. Logic of
today is a fine instrument of pursuing truth and an indispensable insti-
tution ofmathematicalfreedom.Logicliberates mathematics,providing
nonstandard ways of reasoning.
Some modelof settheory is nonstandard if the membership between
the objects of the model differs from that of the originals. In fact, the
nonstandardtools oftodayuse a couple ofset-theoreticmodels simulta-
neously. Boolean valued models reside within the most popular logical
tools.
Boolean valued analysis is a blending of analysis and Boolean val-
ued models which originated and distinguishes itself by ascending and
descending, mixing, cycling hulls, etc.
iv Preface
In this book we show how Boolean valued analysis transforms the
theory of operators in vector lattices. We focus on the recent results
that were not reflected in the monographic literature yet.
InChapter1wecollecttheBooleanvaluedprerequisitesofthefurther
analysis.Chapter2providesthe presentationoftherealsandcomplexes
within Booleanvaluedmodels.In Chapter3 we givethe Booleanvalued
interpretations of order bounded operators with the emphasis on lattice
homomorphisms and disjointness preserving operators. Chapter 4 con-
tains the solutionofthe Wicksteadproblemas wellas other new results
onbandpreservingoperators.Chapter 5 dealswith variousapplications
ofordercontinuousoperatorsto injective Banachlattices,Maharamop-
erators, and related topics.
Adaptationofthe ideasofBooleanvaluedmodelstofunctionalanal-
ysis projects among the most important directions of developing the
synthetic methods of mathematics. This approach yields the new mod-
els of numbers, spaces, and types of equations. The content expands of
allavailabletheorems andalgorithms.The whole methodologyof math-
ematicalresearchisenrichedandrenewed,openingupabsolutelyfantas-
tic opportunities. We can now transform matrices into numbers, embed
function spaces into a straight line, yet having still uncharted vast ter-
ritories of new knowledge. The chase of truth not only leads us close to
the truth we pursue but also enables us to nearly catch up with many
other instances of truth which we were unaware nor even foresaw at the
start of the rally pursuit. That is what we have learned from Boolean
valued analysis.
Quite a long time had passed until the classical functional analysis
occupieditspresentpositionofthelanguageofcontinuousmathematics.
Nowthetimehascomeofthenewpowerfultechnologiesofmodeltheory
inmathematicalanalysis.Notalltheoreticalandappliedmathematicians
havealreadygainedthe importance ofmoderntools andlearnedhow to
use them. However,there is no backwardtraffic in science, and the new
methods are doomed to reside in the realmof mathematics for ever and
they will shortly become as elementary and omnipresent in analysis as
Banach spaces and linear operators.
A. Kusraev
S. Kutateladze
CHAPTER 1
BOOLEAN VALUED REQUISITES
In this chapter we briefly present some prerequisites of the theory of
Boolean valued models. All missing details may be found in Bell [43],
Jech[184],KusraevandKutateladze[248,249],TakeutiandZaring[388].
We mainly keep the notation of [248] and [249].
The most important feature of Boolean valued analysis consists in
comparativeanalysisofthe standardandnonstandard(Booleanvalued)
models under consideration which uses the special technique of ascend-
ing and descending. Moreover, it is often necessary to carry out some
syntactic comparison of formal texts. Therefore, before we launch into
theascendinganddescendingmachinery,wehavetograspacleareridea
of the status of mathematical objects in the framework of a formal set
theory, the construction of a Boolean valued universe, and the way of
assigningtheBooleantruthvaluetoeachsentenceofthelanguageofset
theory.
We use several notations for implication: ⇒ presents the Boolean
operation,=⇒ stands for the logicalconnective, but often in a set theo-
reticformulaweuse→insteadof=⇒indicatingthatthisformulawillbe
interpretedinsome Booleanvaluedmodel. We alsouse ↔,⇔, and⇐⇒
with the similar meaning. The proof-theoretic consequence relation (cid:7)
is applied alongside with the semantic (or model-theoretic) consequence
relation |=.
Observethat,speakingofaformalsettheory,wewillfreely(because
thisisinfactunavoidable)adheretothelevelofrigorwhichiscurrentin
the mainstream of mathematics and introduce abbreviations by means
of the definor, i.e. the assignment operator, := without specifying any
subtleties.
2 Chapter 1. Boolean Valued Requisites
1.1. Zermelo–Fraenkel Set Theory
At present, the most widespread axiomatic foundation for mathe-
matics is Zermelo–Fraenkel set theory. We will briefly recall some of its
concepts, outlining the details we need in the sequel.
1.1.1. The alphabet of Zermelo–Fraenkel theory ZF or ZFC, if the
presence of choice AC is stressed, comprises the symbols of variables;
the parentheses ( and ); the propositional connectives (i.e., the signs
of propositional calculus) ∨, ∧, →, ↔, and ¬; the quantifiers ∀ and ∃;
the equality sign =; and the symbol of the special binary predicate of
containment or membership ∈. In general, the domain of the variables
of ZF is thought as the world or universe of sets. In other words, the
universe of ZF contains nothing but sets. We write x ∈ y rather than
∈(x,y) and say that x is an element of y.
1.1.2. The formulas of ZF are defined by the routine procedure. In
otherwords,theformulasofZFarefinitetextsresultingfromtheatomic
formulasx=y and x∈y, where x andy arevariablesof ZF,by reason-
ably placing parentheses, propositional connectives, and quantifiers
ϕ∨ψ, ϕ∧ψ, ¬ϕ, ϕ→ψ, ϕ↔ψ, (∀x)ϕ, (∃x)ϕ.
So, if ϕ1 and ϕ2 are formulas of ZF and x is a variable then the texts
ϕ1 → ϕ2 and (∃x)(ϕ1 → (∀y)ϕ2)∨ϕ1 are formulas of ZF, whereas
ϕ1∃x and ∀(x∃ϕ1¬ϕ2 are not. We attach the natural meaning to the
termsfreeandbound variables andthe termdomain of a quantifier. For
instance, in the formula (∀x)(x ∈ y) the variable x is bound and the
variable y is free, whereas in the formula (∃y)(x = y) the variable x is
free and y is bound (for it is bounded by a quantifier). Henceforth, in
order to emphasize that the only free variables in a formula ϕ are the
variables x1,...,xn, we write ϕ(x1,...,xn). Sometimes such a formula
is considered as a “function”; in this event, it is convenient to write
ϕ(·,...,·)orϕ=ϕ(x1,...,xn),implying thatϕ(y1,...,yn)isaformula
of ZF obtained by replacing each free occurrence of x by y for k:=
k k
1,...,n.
1.1.3. Studying ZF, it is convenient to use some expressive tools
absent in the formal language. In particular, in the sequel it is worth-
while employing the concepts of class and definable class and also the
corresponding symbols of classifiers like Aϕ:= Aϕ(·):= {x : ϕ(x)} and
Aψ:=Aψ(·,y):={x:ψ(x,y)},whereϕandψareformulasofZFandy is
1.1. Zermelo–Fraenkel Set Theory 3
adistinguishedcollectionofvariables. Ifitisdesirabletoclarifyorelim-
inate the appearing records then we can assume that the use of classes
and classifiers is connected only with the conventionalagreement on in-
troducing abbreviations. This agreement, sometimes called the Church
schema, reads:
z ∈{x:ϕ(x)}↔ϕ(z),
z ∈{x:ψ(x,y)}↔ψ(z,y).
1.1.4.WorkingwithinZF,wewillusesomenotationsthatarewidely
spread in mathematics. We start with the most frequent abbreviations:
x(cid:13)=y:=¬x=y, x∈/ y:=¬x∈y;
(∀x∈y)ϕ(x):=(∀x)(x∈y →ϕ(x));
(∃!z)ϕ(z):=(∃z)ϕ(z)∧((∀x)(∀y)(ϕ(x)∧ϕ(y)→x=y));
(∃x∈y)ϕ(x):=(∃x)(x∈y∧ϕ(x)).
Theempty set∅,the pair{x,y},the singleton{x},the orderedpair
(x,y), and the ordered n-tuple (x1,...,xn) are defined as
∅:={x: x(cid:13)=x};
{x,y}:={z : z =x∨z =y}, {x}:={x,x},
(x,y):={x,{x,y}};
(x1,...,xn):=((x1,...,xn−1),xn).
(cid:2) (cid:3)
Theinclusion⊂,theunion ,theintersection ,thepowersetP(·),
and the universe of sets V are introduced as follows:
x⊂y:=(∀z)(z ∈x→z ∈y);
(cid:4)
x:={z :(∃y ∈x)z ∈y};
(cid:5)
x:={z :(∀y ∈x)z ∈y};
P(x):=“the class of all subsets of x”:={z : z ⊂x};
V:=“the class of all sets”:={x: x=x}.
4 Chapter 1. Boolean Valued Requisites
Notealsothatinthesequelweacceptmorecomplicateddescriptions
in which much is presumed:
Fnc(f):=“f is a function”;
dom(f):=“the domain of f”;
im(f):=“the range of f”;
ϕ(cid:7)ψ:=ϕ→ψ:=“ψ is derivable from ϕ”;
“a class A is a set”:=A∈V:=(∃x)(∀y)(y ∈A↔y ∈x).
Similar simplifications will be used in rendering more complicated for-
mulas without further stipulation. For instance, instead of some rather
involved formulas of ZF we simply write
f :x→y ≡“f is a function from x to y”;
“X is a vector lattice”;
U ∈L∼(X,Y)≡“U is an order bounded linear operator from X to Y.”
1.1.5. In ZFC, we accept the usual axioms and rules of a first-order
theorywithequalitywhichfixthestandardmeansofclassicalreasoning.
Recall the equality axioms:
(1) (∀x)x=x (reflexivity);
(2) (∀x)(∀x)x=y →y =x (symmetry);
(3) (∀x)(∀y)(∀y)x=y∧y =x→x=z (transitivity);
(cid:6)
(4) (∀x)(∀y)(∀u)(∀v) (x=y∧u=v) (cid:7)
→(x∈u→y ∈v) (substitution).
1.1.6. The classical first-order logic CL has the following axiom
schemas (ϕ, ψ, and ω are arbitrary formulas of CL):
(1) ϕ→(ϕ∧ϕ);
(2) (ϕ∧ψ)→(ψ∧ϕ);
(3) (ϕ→ψ)→((ϕ∧ω)→(ψ∧ω));
(4) ((ϕ→ψ)∧(ψ →ω))→(ϕ→ω);
(5) ψ →(ϕ→ψ);
(6) (ϕ∧(ϕ→ψ))→ψ;
(7) ϕ→(ϕ∨ψ);
(8) (ϕ∨ψ)→(ψ∨ϕ);
1.1. Zermelo–Fraenkel Set Theory 5
(9) ((ϕ→ω)∧(ψ →ω))→((ϕ∧ψ)→ω);
(10) ¬ϕ→(ϕ→ψ);
(11) ((ϕ→ψ)∧(ϕ→¬ψ))→¬ϕ;
(12) ϕ∨¬ϕ.
The only rule of inference in CL is modus ponens:
(MP) If ϕ and ϕ→ψ are provable in CL then so is ψ.
1.1.7.Moreover,the followingspecial orproper axiomsareaccepted
in ZFC as a correct formalization of the principles of most mathemati-
cians working with sets:
(1)AxiomofExtensionality. If x and y have the same elements then
x=y:
(∀x)(∀y)(x⊂y∧y ⊂x→x=y).
(cid:2)
(2) Axiom of Union. To each x there exists a set y = x:
(cid:8) (cid:4) (cid:9)
(∀x)(∃y) y = x .
(3) Axiom of Powerset. To each x there exists a set y =P(x):
(∀x)(∃y)(y =P(x)).
(4) Axiom Schema of Replacement. If a class A is a function then
ϕ
to each x there exists a set v =A (x)={A (z): z ∈x}:
ϕ ϕ
(∀x)((∀y)(∀z)(∀u)ϕ(y,z)∧ϕ(y,u)→z =u)
→(∃v)(v ={z :(∃y ∈x)ϕ(y,z)}).
(5) Axiom of Foundation. Each nonempty set has an ∈-minimal
element:
(∀x)(x(cid:13)=∅→(∃y ∈x)(y∩x=∅)).
(6) Axiom of Infinity. There exists an inductive set:
(∃ω)(∅∈ω)∧(∀x∈ω)(x∪{x}∈ω).
(7) Axiom of Choice. Each family of nonempty sets has a choice
function:
(∀F)(∀x)(∀y)((x(cid:13)=∅∧F :x→P(y))
→((∃f)f :x→y∧(∀z ∈x)f(z)∈F(z)).
