Boolean Valued Analysis: Selected Topics

406 Pages·2014·2.161 MB·English

by  Kusraev A.G., Kutateladze S.S.

RUSSIANACADEMYOFSCIENCES 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
BBC 22.16
UDC 517.98
K94
Editor
A.E. Gutman
Reviewers:
S.A. Malyugin and E.Yu. Emelyanov
Editors of the series:
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+406 p.—(Trends in Science: The South of
Russia. A Mathematical Monograph. Issue 6).
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 in two 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
propertiesoforderboundedoperatorsinvectorlattices.Thisvolumeisintendedfor
theclassicalanalystseekingnewpowerfultoolsandforthemodeltheoristinsearch
ofchallengingapplicationsofnonstandardmodelsofsettheory.
Кусраев А. Г., Кутателадзе С. С.
Булевозначный анализ: Избранные темы / отв. ред. А. Е. Гутман.—
Владикавказ:ЮМИВНЦРАНиРСО-А,2014.—iv+406с.—(Итогинауки.
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
методамивихприложениикфункциональномуанализу.
ISBN 978-5-904695-24-8 ©SouthernMathematicalInstitute
VSCRAS&RNO-A,2014
©A.G.Kusraev,2014
©S.S.Kutateladze,2014
PREFACE
Humans definitely feel truth but cannot define truth properly. That
iswhatAlfredTarskiexplainedtousinthe1930s.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 of mathematical freedom. Logic liberates mathematics, providing
nonstandard ways of reasoning.
Some model of set theory is nonstandard if the membership between
the objects of the model differs from that of the originals. In fact, the
nonstandard tools of today use a couple of set-theoretic models 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.Chapter2providesthepresentationoftherealsandcomplexes
within Boolean valued models. In Chapter 3 we give the Boolean valued
interpretations of order bounded operators with the emphasis on lattice
homomorphisms and disjointness preserving operators. Chapter 4 con-
tains the solution of the Wickstead problem as well as other new results
on band preserving operators. Chapter 5 deals with various applications
ofordercontinuousoperatorstoinjectiveBanachlattices,Maharamop-
erators, and related topics.
AdaptationoftheideasofBooleanvaluedmodelstofunctionalanal-
ysisprojectsamongthemostimportantdirectionsofdevelopingthesyn-
thetic methods of mathematics. This approach yields the new models of
numbers,spaces,andtypesofequations.Thecontentexpandsofallavail-
able theorems and algorithms. The whole methodology of mathematical
researchisenrichedandrenewed,openingupabsolutelyfantasticoppor-
tunities. We can now transform matrices into numbers, embed function
spaces into a straight line, yet having still uncharted vast territories of
new knowledge.
Quite a long time had passed until the classical functional analysis
occupieditspresentpositionofthelanguageofcontinuousmathematics.
Nowthetimehascomeofthenewpowerfultechnologiesofmodeltheory
inmathematicalanalysis.Notalltheoreticalandappliedmathematicians
have already gained the importance of modern tools and learned how to
use them. However, there is no backward traffic in science, and the new
methods are doomed to reside in the realm of 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
comparativeanalysisofthestandardandnonstandard(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 logical connective, but often in a set theo-
reticformulaweuse→insteadof=⇒indicatingthatthisformulawillbe
interpreted in some Boolean valued model. We also use ↔, ⇔, and ⇐⇒
with the similar meaning. The proof-theoretic consequence relation (cid:96)
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
formulas x=y and x∈y, where x and y are variables of 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 ϕ and ϕ are formulas of ZF and x is a variable then the texts
1 2
ϕ → ϕ and (∃x)(ϕ → (∀y)ϕ )∨ϕ are formulas of ZF, whereas
1 2 1 2 1
ϕ ∃x and ∀(x∃ϕ ¬ϕ are not. We attach the natural meaning to the
1 1 2
termsfreeandbound variables andthetermdomain 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 x ,...,x , we write ϕ(x ,...,x ). Sometimes such a formula
1 n 1 n
is considered as a “function”; in this event, it is convenient to write
ϕ(·,...,·)orϕ=ϕ(x ,...,x ), implyingthatϕ(y ,...,y )isaformula
1 n 1 n
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 A := A := {x : ϕ(x)} and
ϕ ϕ(·)
A :=A :={x:ψ(x,y)},whereϕandψ areformulasofZFandy is
ψ ψ(·,y)
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 conventional agreement 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:54)=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)).
Theemptyset∅, thepair{x,y}, thesingleton{x}, theorderedpair
(x,y), and the ordered n-tuple (x ,...,x ) are defined as
1 n
∅:={x: x(cid:54)=x};
{x,y}:={z : z =x∨z =y}, {x}:={x,x},
(x,y):={x,{x,y}};
(x ,...,x ):=((x ,...,x ),x ).
1 n 1 n−1 n
(cid:83) (cid:84)
Theinclusion⊂,theunion ,theintersection ,thepowersetP(·),
and the universe of sets V are introduced as follows:
x⊂y:=(∀z)(z ∈x→z ∈y);
(cid:91)
x:={z :(∃y ∈x)z ∈y};
(cid:92)
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:96)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:0)
(4) (∀x)(∀y)(∀u)(∀v) (x=y∧u=v)
(cid:1)
→(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, thefollowingspecial 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:83)
(2) Axiom of Union. To each x there exists a set y = x:
(cid:16) (cid:91) (cid:17)
(∀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:54)=∅→(∃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:54)=∅∧F :x→P(y))
→((∃f)f :x→y∧(∀z ∈x)f(z)∈F(z)).
6 Chapter 1. Boolean Valued Requisites
1.1.8. Grounding on the above axiomatics, we acquire a clear idea
of the class of all sets, the von Neumann universe V. As the initial
object of all constructions we take the empty set. The elementary step
of introducing new sets consists in taking the union of the powersets
of the sets already available. Transfinitely repeating these steps, we
(cid:83)
exhaust the class of all sets. More precisely, we assign V:= V ,
α∈On α
where On is the class of all ordinals and
V :=∅,
0
V :=P(V ),
α+1 α
(cid:91)
V := V (β is a limit ordinal).
β α
α<β
1.1.9. The pair (V,∈) is a standard model of ZFC.
1.2. Boolean Valued Universes
Everywhere below B is a complete Boolean algebra with supremum
(join) ∨, meet (infimum) ∧, complement (·)∗, unit (top) 1, and zero
(bottom) 0. The necessary information on Boolean algebras can be
found in Givant and Halmos [130], Sikorski [365], and Vladimirov [399].
1.2.1. Let B be a complete Boolean algebra. Given an ordinal α,
put
(cid:110)
V(B):= x: Funct(x)∧(∃β)(β <α∧dom(x)
α
(cid:111)
⊂V(B)∧im(x)⊂B) .
β
Thus, in more detail we have
V(B):=∅,
0
V(B) :={x: x is a function with domain in V(B) and range in B};
α+1 α
(cid:91)
V(B):= V(B) (β is a limit ordinal).
α β
β<α
The class (cid:91)
V(B):= V(B)
α
α∈On