Non-Archimedean Analysis on the Extended Hyperreal Line R_d and some transcendence conjectures over field field Q and *Q_omega .

8/14/2019 Non-Archimedean Analysis on the Extended Hyperreal Line *R_d and some transcendence conjectures over field fi

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Non-archimedean analysis on the extended hyperreal

line d and some transcendence conjectures over

field and .

URL: http://arxiv.org/abs/0907.0467

Jaykov Foukzon

Israel Institute of Technology

[email protected]

Abstract. In this paper possible completion of the Robinson non-archimedean

field constructed by Dedekind sections. As interesting example I show how, a

few simple ideas from non-archimedean analysis on the pseudo-ring d gives a

short clear nonstandard reconstruction for the Eulers original proof of theGoldbach-Euler theorem. Given an analytic function of one complex variable

f z, we investigate the arithmetic nature of the values of fat transcendentalpoints.

Contents

Introduction.

1.Some transcendence conjectures over field .

2.Modern nonstandard analysis and non-archimedean analysis on

the extended hyperreal line d.

Chapter I.The classical hyperreals numbers.

I.1.1. The construction non-archimedean field .

I.1.2. The brief nonstandard vocabulary.

I.2. The higher orders of hyper-method.Second order transfer principle.

I.2.1. What are the higher orders of hyper-method?

I.2.2. The higher orders of hyper-method by using countable universes.

I.2.3. Divisibility of hyperintegers.

I.3. The construction non-archimedean pseudo-ring d.

I.3.1. Generalized pseudo-ring of Dedekind hyperreals d.

I.3.2. The topology of d.

I.3.3. Absorption numbers in d.

I.3.3.1. Absorption function and numbers in d.


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I.3.3.2. Special Kinds of Idempotents in d.

I.3.3.3. Types of with a given ab. p. .

I.3.3.4. -Part of with ab. p. 0.I.3.3.5. Multiplicative idempotents.

I.3.3.6. Additive monoid of Dedekind hyperreal integers

d.I.3.5. Pseudo-ring of Wattenberg hyperreal integers d.

I.3.6. External summation of countable and hyperfinite sequences in d.

I.3.7. The construction non-archimedean field d as Dedekind

completion of countable non-standard models of .

I.4. The construction non-archimedean field c.

I.4.1. Completion of ordered group and fields in general by using

"Cauchy pregaps".

I.4.1.1. Totally ordered group and fields

I.4.1.2. Cauchy completion of ordered group and fields.

I.4.2.1. The construction non-archimedean field c by using Cauchy

hypersequence in ancountable field .

I.4.2.2. The construction non-archimedean field c as Cauchy

completion of countable non-standard models of .

Chapter II.Eulers proofs by using non-archimedean analysis on the

pseudo-ring d revisited.

II.1.Eulers original proof of the Goldbach-Euler Theorem revisited.

III. Non-archimedean analysis on the extended hyperreal line d and

transcendence conjectures over field . Proof that e and e is

irrational.

Chapter III.Non-archimedean analysis on the extended hyperreal line dand transcendence conjectures over field .

III.1. Proof that e is #-transcendental and that e and e is irrational.III.2. Nonstandard generalization of the Lindeman Theorem.

List of Notation.

..................................................... the set of infinite natural numbers.................................................the set of infinite hyper real numbers

L L.........................................the set of the limited members of

I I..................................the set of the infinitesimal members of

halox x x I............................................halo (monad) ofx sta......................................................Robinson standard part ofa

z1, . . . ,zn .....................internal polynomials over in n variables


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z1, . . . ,zn .....................internal polynomials over in n variablesz1, . . . ,zn .....................internal polynomials over in n variablesd....................................................Dedekind completion of the field

c.......................................................Cauchy completion of the field

d

supx|x 0

infx|x ...........................0

d,

dd sup inf .........................................................................

WST.........Wattenberg standard part of d, d d (Def.1.3.2.3)

ab. p. ....................................absorption part of d (Def 1.3.3.1.1)

.........................................................................-part ofd d

|b# ...................................................-part of d for a given b

"Arthur stopped at the steep descent into the quarry, froze in his steps, strainingto look down and into the distance, extending his long neck. Redrick joined him.

But he did not look where Arthur was looking.

Right at their feet the road into the quarry began, torn up many years ago by the

treads and wheels of heavy vehicles. To the right was a white steep slope,

cracked by the heat; the next slope was half excavated, and among the rocks

and rubble stood a dredge, its lowered bucket jammed impotently against the

side of the road. And,as was to be expected, there was nothing else to be seen

on the road..."

Arkady and Boris Strugatsky

"Roadside Picnic"

Introduction.1.Some transcendence conjectures over field . In 1873 French

mathematician, Charles Hermite, proved that e is transcendental. Coming as it did100 years after Euler had established the significance of e, this meant that theissue of transcendence was one mathematicians could not afford to ignore.Within

10 years of Hermites breakthrough,his techniques had been extended by

Lindemann and used to add to the list of known transcendental numbers.

Mathematician then tried to prove that other numbers such as e and e aretranscendental too,but these questions were too difficult and so no further

examples emerged till todays time. The transcendence of ehad been provedin1929 by A.O.Gelfond.

Conjecture 1. The numbers e and e are irrational.Conjecture 2. The numbers e and are algebraically independent.

However, the same question with e

and has been answered:Theorem.(Nesterenko, 1996 [22]) The numbers e and are algebraically


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independent.

During of XX th century,a typical question: is whether f is a transcendentalnumber for each algebraic number has been investigated and answered many

authors.Modern result in the case of entire functions satisfying a linear

differential equation provides the strongest results, related with SiegelsE-functions [22],[27].Ref. [22] contains references to the subject before 1998,including Siegel Eand G functions.Theorem.(Siegel C.L.) Suppose that , 1, 2 , . . . , 0.

z n0 zn

1 2 n

Then is a transcendental number for each algebraic number 0.

Given an analytic function of one complex variable fz z, we investigatethe arithmetic nature of the values of fz at transcendental points.

Conjecture 3.Is whether f is a irrational number for giventranscendental number.

Conjecture 4.Is whether f is a transcendental number for giventranscendental number.

In particular we investigate the arithmetic nature of the values of classical

polylogarithms Lisz at transcendental points.The classical polylogarithms

Lisz n1znns

fors 1,2,. . . and |z| 1 with s;z 1; 1, are ubiquitous. The study of thearithmetic nature of their special values is a fascinating subject [35] very few

is known.Several recent investigations concern the values of these functions

at z 1 : these are the values at the positive integers of Riemann zeta function


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s Lisz 1 n11ns

One knows that 3 is irrational [36],and that inInitely many values 2n 1of the zeta function at odd integers are irrational.

Conjecture 4.Is whetherLis is a irrational number for giventranscendental number.

Conjecture 5.Is whetherLis is a transcendental number for giventranscendental number.

2.Modern nonstandard analysis and non-archimedean analysis on the

extended hyperreal line d. Nonstandard analysis, in its early period of

development, shortly after having been established by A. Robinson [1],[4],[5] dealt

mainly with nonstandard extensions of some traditional mathematical structures.The system of its foundations, referred to as "model-theoretic foundations" was

proposed by Robinson and E. Zakon [12]. Their approach was based on the

type-theoretic concept of superstructure VS over some set of individuals Sand itsnonstandard extension (enlargement) VS, usually constructed as a (bounded)ultrapower of the "standard" superstructure VS. They formulated few principlesconcerning the elementary embedding VS VS, enabling the use of methodsof nonstandard analysis without paying much attention to details of construction of

the particular nonstandard extension.

In classical Robinsonian nonstandard analysis we usualy deal only withcompletely internal objects which can defined by internal set theory IST introduced

by E.Nelson [11]. It is known that IST is a conservative extension of ZFC. In IST allthe classical infinite sets, e.g., , , or, acquire new, nonstandard elements (like

"infinite" natural numbers or"infinitesimal" reals). At the same time, the families

x : stx or x : stx of all standard,i.e., "true," natural numbersor reals, respectively, are not sets in IST at all. Thus, for a traditional

mathematician inclined to ascribe to mathematical objects a certain kind of

objective existence or reality, accepting IST would mean confessing that everybody

has lived in confusion, mistakenly having regarded as, e.g., the set just its tiny

part (which is not even a set) and overlooked the rest. Edvard Nelson and KarelHrbc ek have improved this lack by introducing several "nonstandard" set theories

dealing with standard, internal and external sets [13]. Note that in contrast with

early period of development of the nonstandard analysis in latest period many

mathematicians dealing with external and internal set simultaneously,for example

see [14],[15],[16],[17].

Many properties of the standard reals x suitably reinterpreted, can betransfered to the internal hyperreal number system. For example, we have seen

that , like , is a totally ordered field. Also, jast contain the natural number as

a discrete subset with its own characteristic properties, contains the

hypernaturals as the corresponding discrete subset with analogous


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properties.For example, the standard archimedean property

xxyynn|x| |y | n|x| |y| is preserved in non-archimedean field inrespect hypernaturals , i.e. the next property is satisfied

xxyynn|x| |y | n|x| |y|. However, there are many fundamental

properties of do not transfered to

.I. This is the case one of the fundamental supremum propertyof the standard

totally ordered field . It is easy to see that it apper bound property does not

necesarily holds by considering, for example, the (external) set itself which we

ragard as canonically imbedded into hyperreals . This is a non-empty set which

is bounded above (by any of the infinite member in ) but does not have a least

apper bound in . However by using transfer one obtain the next statement [18] :

Weak supremum property for

Every non-empty internalsubset A which has an apper bound in hasa least apper bound in .

This is a problem, because any advanced variant of the analysis on the field is needed more strong fundamental supremum property. At first sight one can

improve this lack by using corresponding external constructions which known as

Dedekind sections and Dedekind completion (see section I.3.).We denote

corresponding Dedekind completion by symbol d. It is clear thatd is

completely external object. But unfortunately d is not iven a non-archimedean

ring but non-archimedean pseudo-ringonly. However this lack does not make

greater difficulties because non-archimedean pseudo-ring d contains

non-archimedean subfield c d such that c c. Here c this is a Cauchy

completion of the non-archimedean field

(see section I.4.).II. This is the case two of the fundamental Peanos induction property

B1 B xx B x 1 B B 1

does not necesarily holds for arbitrary subset B . Therefore (1) istrue for when interpreted in i.e.,

intB 1 B xx B x 1 B B 2

true for provided that we read "B" as "for each internal subsetB of ", i.e. as intB. In general the importance of internal versus external entitiesrests on the fact that each statement that is true for is true for provided its

quantifiers are restricted to the internal entities (subset) of only [18].This is a

problem, because any advanced variant of the analysis on the field is needed

more strong induction property than property (2).In this paper I have improved this

lack by using external construction two different types for operation of exteral


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summation: Ext n

qn, #Ext n

qn# and two different types for operation of

exteral multiplication: Ext n

qn, #Ext n

qn# for arbitrary countable sequences

such as qn : and qn# : d.As interesting example I show how, this external constructions from

non-archimedean analysis on the pseudo-ring d gives a short and clear

nonstandard reconstruction for the Eulers original proof of the Goldbach-Euler

theorem.

I.The classical hyperreals numbers.

I.1.1.The construction non-archimedean field .Let denote the ring of real valued sequences with the usual pointwise

operations.Ifx is a real number we let sx denote the constant sequence,sx x forall n. The function sending x to sx is a one-to-one ring homomorphism,providing anembedding of into . In the following, wherever it is not too confusing we will not

distinguish between x and the constant function sx, leaving the reader to deriveintent from context. The ring has additive identity 0 and multiplicative identity 1.

is not a field because ifris any sequence having 0 in its range it can have nomultiplicative inverse. There are lots of zero divisors in .

We need several definitions now. Generally, for any set S, PS denotes the set

of all subsets of S. It is called the power set of S. Also, a subset of will be calledcofinite if it contains all but finitely many members of . The symbol denotes the

empty set. A partition of a set S is a decomposition of Sinto a union of sets, anypair of which have no elements in common.

Definition.1.1.1. An ultrafilterH over is a family of sets for which:

(i) H P, H.(ii) Any intersection of finitely many members of H is in H.

(iii) A ,B H A B H.(iv) IfV1, . . . , Vn is any finite partition of then H contains exactlyone of the Vi.If, further,

(v) H contains every cofinite subset of .

the ultrafilter is called free.

If an ultrafilter on contains a finite set then it contains a one-point set, and is

nothing more than the family of all subsets of containing that point. So if an

ultrafilter is not free it must be of this type, and is called a principal ultrafilter.

The existence of a free ultrafilter containing any given infinite subset of is

implied by the Axiom of Choice.

Remark 1.1.1. Suppose that x X. An ultrafilter denoted prinXx X

consisting of all subsets S Xwhich contain x, and called the principalultrafiltergenerated by x.


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Proposition 1.1.1. If an ultrafilter on Xcontains a finite set S X, then isprincipal.

Proof: It is enough to show contains x for some x S. If not, then contains the complement X\x for every x S, and therefore also the finite

intersection xSX\x X\S, which contradicts the fact that S .It follows that nonprincipal ultrafilters can exist only on infinite sets X, and thatevery cofinite subset of X(complement of a finite set) belongs to such anultrafilter.

Remark 1. 1. 2. Our construction below depends on the use of a free-not a

principal-ultrafilter.

We are going to be using conditions on sequences and sets to define subsets of

. We introduce a convenient shorthand for the usual set builder notation. If Pis aproperty that can be true or false for natural numbers we use P to denote

n |Pn is true . This notation will only be employed during a discussion todecide if the set of natural numbers defined by Pis in H, or not. For example, ifs, tis a pair of sequences in we define three sets of integers For example, if s, tis apair of sequences in Swe define three sets of integers

s t, s t, s t. 1. 1

Since these three sets partition , exactly one of them is in H, and we

declare s twhen s t H.Lemma 1.1.1. is an equivalence relation on . We denote the equivalenceclass of any sequence s under this relation by s. Define for each r thesequence r by

r 0 iffrn 0

rn1 iffrn 0. 1. 2

Lemma 1.1.2. (a) There is at most one constant sequence in any class r.(b) 0 is an ideal in so /0 is a commutative ring with identity [1].

(c) Consequently r r 0 r t|t 0 for all r .(d) Ifr 0 then r r 1. So r1 r.From Lemma 1.1.2., we conclude that , defined to be /0, is a field

containing an embedded image of as a subfield. 0 is a maximal ideal in .

Definition.1.1.2.This quotient ring is called the field ofclassical hyperreal

numbers.

We declare s t provided s t H.Recall that any field with a linear order is called an ordered field provided


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(i) x y 0 whenever x,y 0(ii) x y 0 whenever x,y 0(iii) x z y z whenever x yTheorem 1.1.3. (a) The relation given above is a linear order on , and makes

into an ordered field. As with any ordered field, we define |x| forx

tobe x orx, whichever is nonnegative.(b) Ifx,y are real then x y if and only if x y. So the ring morphismof into is also an order isomorphism onto its image in .

Because of this last theorem and the essential uniqueness of the real numbers

it is common to identify the embedded image of in with itself. Though

obviously circular, one does something similar when identifying with its

isomorphic image in , and itself with the corresponding subset of . This kind

of notational simplification usually does not cause problems.

Now we get to the ideas that prompted the construction. Define the sequence rby rn n 11 . For every positive integer k, r k1 H. So 0 r 1/k. Wehave found a positive hyperreal smaller than (the embedded image of) any real

number. This is our first nontrivial infinitesimal number. The sequence r is given byr n n 1. So r1 r kfor every positive integer k. r1 is a hyperreal largerthan any real number.

I.1.2.The brief nonstandard vocabulary.

Definition.1.1.2.1. We call a member x -limited if there are membersa, b with a x b.We will use L L to indicate the limited members of . x is called-unlimited if it not -limited.

These terms are preferred to finite and infinite,

which are reserved for concepts related to cardinality.

Definition.1.1.2.2. Ifx,y and x y we use x,y to denotet |x t y.This set is called a closed hyperinterval. Open and half-open hyperintervals

are defined and denoted similarly.

Definition.1.1.2.3. A set S is called hyperbounded if there are membersx,y of for which Sis a subset of the hyperinterval x,y.Abusing standard vocabulary for ordered sets, S is called bounded if x and ycan be chosen to be limited members of x and y could, in fact, be chosento be real ifSis bounded.The vocabulary of bounded or hyperbounded above and below can be used.

Definition.1.1.2.3. We call a member x infinitesimal if |x| a for everypositive a . We write x 0 iffx is infinitesimal.


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The only real infinitesimal is obviously 0.

We will use I I to indicate the infinitesimal members of .

Definition.1.1.2.4. A memberx is called appreciable if it is limited but notinfinitesimal.

Definition.1.1.2.5. Hyperreals x and y are said to have appreciable separation if|x y| is appreciable.We will be working with various subsets Sof and adopt the followingconvention: S S\L x S|x L. These are the unlimited members ofS, if any.Definition.1.1.2.6. (a) We say two hyperreals x,y are infinitesimally close orhave infinitesimal separation if |x y| I.We use the notation x y to indicate that x and y are infinitesimally close.(b) They have limited separation if |x y| L.(c) Otherwise they are said to have unlimited separation.

We define the halo of x by halox x I. There can be at most one realnumber in any halo. Whenever halox is nonempty we define the shadowofx, denoted shadx, to be that unique real number.The galaxy of x is defined to be galx x L. galx is the set of hyperrealnumbers a limited distance away from x. So ifx is limited galx L.If n is any fixed positive integer we define n to be the set of equivalence

classes of sequences in n under the equivalence relation x y exactly whenx y H.Definition.1.1.2.7. We call the set ofclassical hypernatural or A. Robinsons

hypernatural numbers,

the set ofclassical infinite hypernatural or A. Robinsons infinite hypernatural numbers, the set ofclassical infinite

hyperreal or A. Robinsons infinite hyperreal numbers, the set ofclassical

hyperintegers or A. Robinsons hyperintegers, and the set ofclassical

hyperrational numbers or A. Robinsons hyperrational numbers.

Theorem 1.1.2.1. is not Dedekind complete.

(hint: is bounded above by the member t , where t is the sequencegiven by tn n for all n . But can have no least upper bound: if n cfor all n then n c 1 for all n .As another example consider I. This set is (very) bounded, but has no least

upper bound.)

Theorem 1.1.2.2. For every r there is unique n with n r n 1.

I.2.The higher orders of hyper-method.Second order

transfer principle.

I.2.1.What are the higher orders of hyper-method?


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Usual nonstandard analysis essentially consists only of two fundamental tools:

the (first order) star-map 1 and the (first order) transfer principle.In most applications, a third fundamental tool is also considered, namely the

saturation property.

Definition.1.2.1.1. Any universe U is a nonempty collection of"standardmathematical objects"that is closed under subsets, i.e. a A U a Uand closed under the basic mathematical operations.Precisely, whenever

A,B U, we require that also the union A B, the intersection A B, theset-difference A\B the ordered pairA,B, the Cartesian product A B, thepowerset PA a|a A, the function-set BA f | f : A B all belong

to U. A universe U is also assumed to contain (copies of) all

sets of numbers , , , , U, and to be transitive, i.e. members ofmembers ofU belong to U or in formulae: a A U a U.

The notion of "standardmathematical object" includes all objects used in the

ordinary practice of mathematics, namely: numbers, sets, functions, relations,

ordered tuples, Cartesian products, etc. It is well-known that all these notions

can be defined as sets and formalized in the foundational framework

of Zermelo-Fraenkel axiomatic set theory ZFC.

From standard assumption: ConZFC and Gdels completeness theoremone obtain that ZFC has a model M. A model Mof set theory is called standardif the element relation M of the model Mis the actual element relation restricted to the model M, i.e.M |M . A model is called transitive when it isstandard and the base class is a transitive class of sets. A model of set theory

is often assumed to be transitive unless it is explicitly stated that it is

non-standard. Inner models are transitive,transitive models are standard, and

standard models are well-founded.

The assumption that there exists a standard model of ZFC (in a given universe)

is stronger than the assumption that there exists a model. In fact, if there is a

standard model, then there is a smallest standard model called the minimal

model contained in all standard models. The minimal model contains nostandard model (as it is minimal) but (assuming the consistency ofZFC) it

contains some model of ZFC by the Gdel completeness theorem. This model

is necessarily not well founded otherwise its Mostowski collapse would be a

standard model. (It is not well founded as a relation in the universe, though it

satisfies the axiom of foundation so is "internally" well founded.

Being well founded is not an absolute property[2].) In particular in the minimal

model there is a model of ZFC but there is no standard model of ZFC.

By the theorem of Lwenheim-Skolem, we can choose transitive models M ofZFC of countable cardinality.


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Remark 1.2.1.1.In ZFC, an ordered pair a, b is defined as the Kuratowski paira, a, b; an n-tuple is inductively defined by a1, . . . , an, an1 a1, . . . , an , an1 ; an n-place relation R on A is identied with the set R An ofn-tuples that satisfy it; a function f : A B is identied with its grapha, b A B|b fa.As for numbers, complex numbers are defined as ordered pairs of

real numbers, and the real numbers are defined as equivalence classes

of suitable sets of rational numbers namely, Dedekind cuts or Cauchy

sequences.

The rational numbers are a suitable quotient /, and the integers are

in turn a suitable quotient /. The natural numbers of ZFC are defined as

the set of von Neumann naturals: 0 0 and n 1 n (so that each natural

number n 0,1,. . . , n 1 is identified with the set of its predecessors.)

Each countable model M ofZFC contains countable model of thereal numbers . Every element x defines a Dedekind cut:x q |q x q |q x. We therefore get a order preserving map

fp : 1.2.1

and which respects addition and multiplication.

We address the question what is the possible range of fp?

Proposition 1.2.1.1. Choose an arbitrary subset . Then there is amodel such that fp . Moreover, the cardinality of canbe chosen to coincide with , if is infinite.

Proof. Choose . For each chooseq1 q2 . . . p1 p2 with

nlim qn

nlim pn .

We add to the axioms of the following axioms:

ekk qk e pk 1.2.2

Again is a model for each finite subset of these axioms,so that the

compactness theorem implies the existence of as required,where

the cardinality of can be chosen to be the cardinality of the set of

axioms, i.e. of , if is infinite. Note that by construction fpe .


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Remark 1.2.1.2. It follows by the theorem of Lwenheim-Skolem,that

for each countable subset we can find a countable model of such that the image of fp contains this subset.Note,on the other hand,that the image will only be countable, so that the different models

will have very different ranges.

Hyper-Tool # 1: FIRST ORDER STAR-MAP.

Definition.1.2.1.2. The first orderstar-map is a function 1 : U V1between two universes that associates to each object A U its first order

hyper-extension (orfirst ordernon-standard extension)

1

A V1. It is alsoassumed that 1 n n for all natural numbers n , and that the propernesscondition 1 holds.Remark 1.2.1.1. It is customary to call standard any object A U in thedomain of the first order star-map 1 , and first order nonstandard any objectB V1 in the codomain. The adjective standard is also often used in theliterature for first order hyper-extensions 1A V1.Hyper-Tool # 2: SECOND ORDER STAR-MAP.

Definition.1.2.1.3. The second orderstar-map is a function 2 : V1 V2between two universes that associates to each object A V1 its second

orderhyper-extension (orsecond ordernon-standard extension) 2A V2.It is also assumed that 2N Nfor all hyper natural numbers N 1 , andthat the properness condition 2 1 holds.Hyper-Tool # 3: FIRST ORDER TRANSFER PRINCIPLE.

Definition.1.2.1.4. Let Pa1, . . . , an be a property of the standard objectsa1, . . . , an U expressed as an "elementary sentence". Then Pa1, . . . , an istrue if and only if corresponding sentence 1Pc1, . . . , cn is true about thecorresponding hyper-extensions 1 a1, . . . , 1 an V1. That is:

Pa1, . . . , an 1P1 a1, . . . , 1 an .

In particularPa1, . . . , an is true if and only if the same sentence Pc1, . . . ,cn is true about the corresponding hyper-extensions 1 a1, . . . , 1 an V1. That is:Pa1, . . . , an P1 a1, . . . , 1 an .Hyper-Tool # 4: SECOND ORDER TRANSFER PRINCIPLE.

Definition.1.2.1.5.Let 1P1 a1, . . . , 1 an be a property of the first ordernon-standard objects 1 a1, . . . , 1 an V1 expressed as an "elementarysentence".


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I.2.2.The higher orders of hyper-method by using

countable universes.

Definition.1.2.2.1. Any countable universe U is a nonempty countable

collection of"standard mathematical objects" that is closed under subsets,

i.e. a A U a U and closed under the basic mathematical operations.Precisely, whenever

A,B U, we require that also the union A B, the intersection A B, theset-difference A\B the ordered pairA,B, the Cartesian product A B, thepowerset PA a|a A, the function-set BA f | f : A B all belong to

U. A countable universe U is also assumed to contain (copies of) all

sets of numbers , , U, , U, and to be transitive, i.e. members ofmembers ofU belong to U or in formulae: a A U a U.Remark 1.2.2.1.In any countable model M ofZFC, an ordered paira, b isdefined as the Kuratowski pair

a, a, b; an n-tuple is inductively defined by a1, . . . , an, an1

a1, . . . , an , an1 ; an n-place relation R on A is identied with the countable setR An ofn-tuples that satisfy it; a function f : A B is identied with its grapha, b A B|b fa.As for numbers, complex numbers are defined as ordered pairs of

real numbers, and the real numbers are defined as countable set of

countable equivalence classes of suitable sets of rational numbers namely,

Dedekind cuts or Cauchy sequences.

The rational numbers are a suitable quotient /, and the integers are

in turn a suitable quotient /. The natural numbers of ZFC are defined as

the set of von Neumann naturals: 0

0

and n

1

n (so that each naturalnumber n 0,1,. . . , n 1 is identified with the set of its predecessors.)

I.2.3.Divisibility of hyperintegers.

Definition.1.2.3.1.Ifn and dare hypernaturals,i.e. n, d or hyperintegers,


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i.e. n, d and d 0, then n is divisible by dprovided n d kfor somehyperinteger k. Alternatively, we say:1.n is a multiple of d,2.dis a factor ofn,

3.dis a divisor ofn,4.ddivides n (denoted with d | n).Theorem 1.2.3.1.Transitivity of Divisibility.

For all a, b, c , ifa|b and b|c, then a|c.Theorem 1.2.3.2.Every positive hyperinteger greater than 1

is divisible by a hyperprime number.

Definition.1.2.3.2.Given any integern 1, thestandard factored form of n is an expression ofn

k1m pk

ek,

where m is a positive hyperinteger, p1,p2, . . . ,pm arehyperprime numbers with p1 p2 . . . pm ande1, e2, . . . , em are positive hyperintegers.Theorem 1.2.3.3.Given any hyperintegern 1, there existpositive hyperintegerm, hyperprime numbers p1,p2, . . . ,pmand positive hyperintegers e1, e2, . . . , em with n k1

m pkek.

Theorem 1.2.3.1. (i) Every pair of elements m, n has a highestcommon factord s m t n for some s, t .(ii) For every pair of elements a, d dividend a and divisord, with d 0there exist unique integers q and rsuch that a q d rand 0 r |d|.

Definition.1.2.3.2. Suppose that a q d rand 0 r |d|. We call dthequotient and rthe remainder.

Redrick, squinting his swollen eyes against the blinding light, silently watchedhim go. He was cool and calm, he knew what was about to happen, and he

knew that he would not watch,but it was still all right to watch, and he did,

feeling nothing in particular,except that deep inside a little worm started

wriggling around and twisting its sharp head in his gut.

Arkady and Boris Strugatsky

"Roadside Picnic"

I.3.The construction non-archimedean

pseudo-ringd.


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I.3.1.Generalized pseudo-ring Dedekind

hyperreals d.From Theorem 1.2.1. above we knov that: is not Dedekind complete.

For example, 0 and are bounded subsets of which have no

suprema or infima in .

Possible standard completion of the field can be constructed by

Dedekind sections [23],[24]. In [24] Wattenberg constructed the Dedekind

completion of a nonstandard model of the real numbers and applied the

construction to obtain certain kinds of special measures on the set of integers.

Thus was established that the Dedekind completion d of the field is a

structure of interest not for its own sake only and we establish further importent

applications here. Importent concept was introduce Gonshor [23] is that ofthe absorption numberof an element a d which, roughly speaking,measures the degree to which the cancellation law a b a c b cfails fora.

More general construction well known from topoi theory [10].

Definition 1.3.1.1. A Dedekind hyperreal d is a pairU, V P P satisfying the next conditions:1.xyx U y V.2. U

V .

3.xx U yy V x y.4. xx V yy V y x.5. xyx y x U y V.

Remark. The monad of , the set x

| x is denoted: .

Monad 0 is denoted: I. Supremum of I is denoted: d.

Let A be a subset of is bounded or hyperbounded above thensupA exists in d.

Example. (i) d sup d\, (ii) d sup I d\.Remark. Anfortunately the set d inherits some but by no means all

of the algebraic structure on . For example,d is not a group with

respect to addition since if x d y denotes the addition ind then:

d d d d d 0 d d

Thus d is not iven a ring but pseudo-ring only. Thus, one must

proceed somewhat cautiously. In this section more details than iscustomary will be included in proofs because some standard properties


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which at first glance appear clear often at second glance reveal themselves

to be false in d.

We shall briefly remind a way Dedekinds constructions of a pseudo-fieldd.

Definition 1.3.1.2.a.(General) Suppose is a total ordering on X.Then A,B is said to be a Dedekind cut of X, ,if and only if:1. A and B are nonempty subsets of X.2. A B X.3. For each x in A and each y in B,x y.A is the left-handpart of the cut A,B and B is the riht-handpartof the cut A,B. We denote the cut as x A|B or simple x A.

The cut x A|B is less than or equal to the cut y C|D ifA C.We write x y ifx is less than or equal to y and we write x y ifx y and x y.Definition 1.3.2.b. c Xis said to be a cut element ofA,B if andonly if either:

(i) c is in A and x c y for each x in A and each y in B, or(ii) c is in B and x c y for each x in A and each y in B.Definition 1.3.2.c.X, is said to be Dedekind complete if and only ifeach Dedekind cut of X, ,has a cut element.Example. The following theorem is well-known.

Theorem. , is Dedekind complete, and for each Dedekind cut

A,B,of, ifrand s are cut elements of A,B, then r s.Making a semantic leap, we now answer the question "what is a

Dedekind hyperreal number ?"

Definition 1.3.2.d. A Dedekind hyperreal number is a cut in .d is the class of all Dedekind hyperreal numbers x A|B (x A).We will show that in a natural way d is a complete ordered

generalized pseudo-ring containing .

Before spelling out what this means, here are some examples of cuts.

(i)A|B r

| r 1 r

| r 1 .

(ii)

A|B r

|r 0 r2 2 r

|r 0 r2 2 .

(iii)

A|B r

| r r

| r , where \.

(iv)

A|B r

|r 0 r I r

r |r 0 r \ I .


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(v)

A|B r

|r 0 r I r

|r 0 r

\I .

Remark. It is convenient to say that A|B d is a rational (hyperrational)

cut in if it is like the cut in examples (i),(iii): fore some fixed rational(hyperrational) numberc ,A is the set of all hyperrational rsuch thatr c while B is the rest of .

The B-set of a rational (hyperrational) cut contains a smollest c , andconversaly ifA|B is a cut in and B contains a smollest element c then A|Bis a rational or hyperrational cut at c. We write c for the rational hyperrationalcut at c. This lets us think of d by identifying c with c.Remark. It is convenient to say that:

(1) A|B d is an standard cut in if it is like the cut in examples (i)-(ii):fore some cut A |B the next equality is satisfied:A|B A |B , i.e. A-setof a cut is an standard set.

(2) A|B d is an internal cut ornonstandard cut in if it is like the cut inexample (iii), i.e. A-set of a cut is an internal nonstandard set.(3) A|B d is an external cut in if it is like the cut in examples (iv)-(v),i.e. A-set of a cut is an external set.There is an order relation on cuts that fairly cries out for attention.Definition 1.3.2.e. The cut x A|B is less than or equal to the cut y C|DifA C.We write x y ifx is less than or equal to y and we write x y if

x y and x y. Ifx A|B is less than y C|D then A Cand A C, sothere is some c0 C\A. Sinse the A-set of a cut contains no largest element,there is also a c1 Cwith c0 c1. All the hyperrational numbers c withc0 c c1 belong to C\A.Remark. The property distinguishing d from

and from and which is

the bottom of every significant theorem about d involves upper bounds and

least upper bounds or equivalently,lower bounds and gretest lower bounds.

Definition 1.3.2.f. M d is an upper bound for a set S d if each s Ssatisfies s M. We also say that the set Sis bounded above by Miff

M L

We also say that the set Sis hyperbounded above iffM L

,i.e.|M| \.Definition 1.3.2.g. An upper bound for Sthat is less than all other upperbound forSis a least upper bound forS.The concept of a pseudo-ringoriginally was introduced by

E. M.Patterson [21].Briefly,Pattersons pseudo-ring is an algebraic

system consisting of an additive abelian group A, a distinguished

subgroup A ofA, and a multiplication operation AA A under

which A is a ring and A a left A-module.For convenience, we denote

the pseudo-ring by A,A.

Definition 1.3.1.2.h.Generalized pseudo-ring (m-pseudo-ring) is analgebraic system consisting of an abelian semigroup As (or abelian monoid


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Am),a distinguished subgroup As ofAs (or a distinguished subgroup Am

ofAm),and a multiplication operation As

As As (Am

Am Am) under

which As (Am

) is a ring and As (Am) a left As-module (Am

-module).

For convenience,we denote the generalized pseudo-ring by s As,As.

Pseudo-field is an algebraic system consisting of an abelian semigroup As,a distinguished subgroups As

As# ofAs and a multiplication operations

As

As Asand AsAs

As under which As is a ring,As

# is a field and Asis a vector spase over field As

# .

Definition 1.3.1.2. A Dedekind cut in is a subset of thehyperrational numbers that satisfies these properties:

1. is not empty.

2. \ is not empty.

3. contains no greatest element

4. Forx,y , ifx and y x, then y as well.Definition 1.3.1.3. A Dedekind hyperreal number d is a Dedekindcut in . We denote the set of all Dedekind hyperreal numbers byd and we order them by set-theoretic inclusion, that is to

say, for any , d, if and only if where theinclusion is strict. We further define as real numbers

if and are equal as sets. As usual, we write if

or .

Definition 1.3.1.4. A hyperreal number is said to be Dedekind

hyperirrationalif \ contains no least element.

Theorem 1.3.1.1. Every nonempty subset A d of Dedekindhyperreal numbers that is bounded (hyperbounded) above has a least

upper bound.

Proof. Let A be a nonempty set of hyperreal numbers, such that forevery A we have that for some real number d.

Now define the set supA A

. We must show that this set is a

Dedekind hyperreal number. This amounts to checking the four conditions

of a Dedekind cut. supA is clearly not empty, for it is the nonempty unionof nonempty sets. Because is a Dedekind hyperreal number, there is some

hyperrational x that is not in . Since every A is a subset of ,x is notin any , so x supA either. Thus, \supA is nonempty. IfsupA had a greatestelement g , then g for some A. Then gwould be a greatestelement of, but is a Dedekind hyperreal number, so by contrapositive law,

supA has no greatest element. Lastly, ifx and x supA, then x forsome , so given any y , y x because is a Dedekind hyperreal numbery whence y supA. Thus supA, is a Dedekind hyperreal number.

Trivially,supA is an upper bound ofA, for every A, supA. It now sufficesto prove that supA , because was an arbitrary upper bound. But this is easy,because every x supA,x is an element of for some A, so because


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,x . Thus, supA is the least upper bound of A.Definition 1.3.1.5. Given two Dedekind hyperreal numbers and we define:

1.The additive identity (zero cut) 0 d , denoted 0, is

0 d x

| x 0 .

2.The multiplicative identity 1 d , denoted 1, is1 d x

| x 1 .

3. Addition d of and denoted is

x y| x ,y .

It is easy to see that d 0 d 0 d for all d.

It is easy to see that d is a cut in and d d .

Another fundamental property of cut addition is associativity:

d d d d .

This follows from the corresponding property of .

4.The opposite d of, denoted , is x

| x , x is not the least element of

\ .

5.Remark. We also say that the opposite of is the additive inverseof denoted iff the next equality is satisfied: 0.

6.Remark. It is easy to see that for all internal cut Int the opposite Int

is the additive inverse ofInt, i.e. Int Int 0.

7.Example. (External cut Xwithout additive inverse X) For any x,y wedenote: x r r x

r , y r r x

r .

Let us consider two Dedekind hyperreal numbers Xand Ydefined as:

X x| x

x ,Z x|z 0.It is easy to see that is no exist cut Ysuch that: X Y Z.Proof. Suppose that cut Ysuch that: X Y Zexist. It is easy to check thatyy Y y . Suppose that y Y, then xx X x y Z,i.e.x x y 0. Hence x y x, i.e. y . It is easy to checkthat Z X Y. Ifx Xand y Ythen x and y , hence x y 1, i.e.1 X Y. Thus Z X Y. This is a contradiction.8.We say that the cut is positive if0 or negative if 0.

The absolute value of , denoted ||, is || , if 0 and || , if 09.If, 0 then multiplication

d

of and denoted is

z

| z x y for some x ,y with x,y 0 .

In general, 0 if 0 or 0,

|| || if 0, 0 or 0, 0, || || if 0, 0, or 0, 0.10. The cut order enjois on d the standard additional properties of:

(i) transitivity: .(ii) trichotomy: eizer , or but only one of the three

things is true.

(iii) translation: d d .11.By definition above, this is what we mean when we say that d is an


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ordered pseudo-ring orordered pseudo-field.

Remark. We embed in d in the standard way [24].If thecorresponding element, #, of d is

# x |x

Lemma 1.3.1.1.[24].

(i) Addition d is commutative and associative ind.

(ii) : d 0 d .(iii) , : # d

# d #

.

Proof. (i) Is clear from definitions.

(ii) Suppose a . Since a has no greatest element bb a b .

Thus a b 0 d and a a b b 0 d .(iii) (a) # d

# d # is clear since:

x y x y .

(b) Suppose x . Thus x

2 and

x2

.

So one obtain x x

2

x2

# d #,

d #

# d #.

Notice, here again something is lost going from to d since a doesnot imply since 0 d but 0 d d d d.

Lemma 1.3.1.2.[24].(i) d a linear ordering on

d, which extends the usual ordering on.

(ii) .(iii) .(iv) is dense in d. That is if in d there is an a then

a# .

Lemma 1.3.1.3.[24].

(i) If

then

d #

#

.(ii) d d .(iii) d d d d .

(iv) d d d d d d

.

(v) a : a#

d d

d a# d .

(vi) d d d 0 d .

Lemma 1.3.1.4.[24].

(i) a, b : a b# a# d b#.


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(ii) Multiplication is associative and commutative:

d d , d d d d .

(iii) 1 d d ; 1 d d d , where 1 d 1 #.

(iv) || d || || d ||.

(v) 0 0 0 d d d d d .

(vi) 0 d d d , 0 d d d

d d d

.

Proof.(v) Clearly d d d d d .Suppose d d d d . Hence:d ab ac, where a, a , b , c .Without loss of generality we may assume a a. Hence:d ab ac a b a c a b c d d .

Definition 1.3.1.6. Suppose d, 0 d then 1 d is defined

as follows:

(i) 0 d d : 1 d infa1 |a ,

(ii) d d 0 : 1 d d d

1 d .

Lemma 1.3.1.5.[24].

(i) a : a# 1 d a1 #.

(ii) 1 1

.

(iii) 0 d d d 1 d d

1 d .

(iv) 0 d d 0 d d 1 d

1 d d 1

(v) a : a 0 # 1 d d

1 d # d 1 d .

(vi) d 1 d d 1 d .

Lemma 1. 3. 1. 5.Suppose that a , a 0, , d. Then

a# d d a# d d a

# d .

Proof. Clearly a# d d a# d d a

# d .

a# 1

a# d d a# d

a# 1 a# d d a# 1 a# d

d . Thus a#

1a# d d a

# d d andone obtain a# d d a

# d a# d d .

Lemma 1.3.1.4. (General Strong Approximation Property).

IfA is a nonempty subset of d which is bounded from above, thensupA is the unique number such that:(i) supA is an upper bound for A and(ii) for any supA there exists x A such that x supA.Proof. If not, then is an upper bound ofA less than the least upper

bound supA, which is a contradiction.Lemma 1.3.1.5.Let A and B be nonempty subsets of d and


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C a b : a A, b B.IfA and B are bounded or hyperboundedfrom above,hence supA and supB exist, then s-supC exist and

supC supA supB.

Proof.Suppose c supA supB. From Lemma 1.3.1.2.(iv) isdense in d. So there is exists x such that c x# supA supB.Suppose that , and # supA, # supB. From Lemma 1.3.1.4(General Strong Approximation Property)one obtain there is exists

a A, b B such that # a supA, # b supB. Supposex# # #. Thus one obtain:

# # # x#

2 # a supA

and

# # # x#

2 # b supB.

So one obtain

x# # # # x#

2 #

# # x#

2 # #

a b supA supB.

But a b C,hence by using Lemma 1.3.1.4 one obtain thatsupC supA supB.

Theorem 1.3.1.2. Let A and B be nonempty subsets of d and

C a b : a A, b B.IfA and B are bounded or hyperboundedfrom above,hence supA and supB exist, then s-supC exist and

supC supA supB. 1.3.3.1


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Proof.Suppose c supA supB. From Lemma 1.3.1.2.(iv) isdense in d. So there is exists x such that c x# supA supB.Suppose that , and # supA, # supB. From Lemma 1.3.1.4(General Strong Approximation Property)one obtain there is exists

a A, b B such that #

a supA, #

b supB. Supposex# # #. Thus one obtain:

# # # x#

2 # a supA

and

# # # x#

2 # b supB.

So one obtain

x# # # # x#

2 #

# # x#

2 # #

a b supA supB.

But a b C,hence by using Lemma 1.3.1.4 one obtain thatsupC supA supB.

Theorem 1.3.1.3.Suppose that S is a non-empty subset of d which is

bounded or hyperbounded from above,i.e. supS exist and suppose that

, 0.Then

xS

sup # x # xS

sup x # sup S. 1.3.3.2

Proof.Let B s-sup S.Then B is the smallest number such that, forany x S,x B.Let T # x|x S. Since # 0, # x # B for anyx S.Hence T is bounded or hyperbounded above by # B. Hence

T has a supremum CT

s-sup T. Now we have to pruve that CT

#

B

# sup S. Since # B # sup S is an apper bound for Tand Cis the


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smollest apper bound for T, CT # B. Now we repeat the argument abovewith the roles of S and T reversed. We know that CT is the smallest numbersuch that, for any y T,y CT. Since 0 it follows that#

1 y # 1 CT for any y T.But S #

1 y|y T . Hence

# 1

CT is an apper bound for S.But B is a supremum for S.HenceB # 1 CT and # B CT. We have shown that CT # B and alsothat # B CT. Thus # B CT.

Theorem 1.3.1.4. Suppose that 0, d, d. Then

# d d # d d

# d . 1.3.3.3

Proof.Let us consider any two sets S and S such that: supS , supS . Thus by using Theorem 1.3.1.3 andTheorem 1.3.1.2 one obtain:

# d d # d supS S

sup# d S S sup# d S

# d S

sup# d S sup# d S

# d supS # d supS .

Theorem 1.3.1.5. Suppose that

,

0,

,

d. Then

# d # d 1 d d |

# | d # d |

# | d . 1.3.3.4

Proof.Let us consider any set S such that: supS . Thus byusing Theorem 1.3.1.3, Theorem 1.3.1.2 and Lemma 1.3.1.3 (v) one obtain:


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# d # d |

# | d 1 d d # d

|# | d d # d d

|# | d d # d |

# | d d

|# | d 1 d d # d |

# | d 1 d d

# d # d

# d .

I.3.2.The topology of d. Wattenberg standard part.Fortunately topologically, d has many properties strongly reminiscent

of itself. We proceed as follows [24].

Definition 1.3.2.1.

(i) , d u| d u d ,

(ii) , d u| d u d .

Definition 1.3.2.2.[24].Suppose U d. Then Uis open if and only

if for every u U, d d d u d such thatu , d U.

Remark.1.3.2.1.[24]. Notice this is not equivalent to:

uuU0 u , u d U .

Lemma 1.3.2.1.[24].

(i) is dense in d.

(ii) d\ is dense in d.

Lemma 1.3.2.2.[24]. Suppose A d. Then A is closed if andonly if:

(i) EE A Ebounded above implies supE A, and(ii) EE A Ebounded below implies infE A.

Proposition 1.3.2.1.[24].

(i) d is connected.

(ii) For d ind set , d is compact.

(iii) Suppose A d. Then A is compact if andonly ifA is closed and bounded.

(iv) d is normal.


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(v) The map d is continuous.

(vi) The map 1 d is continuous.

(vii) The maps , d and , d

are not continuous.

Definition 1.3.2.3.[24].(Wattenberg Standard Part)

(i) Suppose d, d d . Then there is a unique

standard x called WST, such that x d, d d ,

(ii) d implies WST WST,(iii) the map WST : d is continuous,(iv) WST d WST WST,(v) WST d WST WST,(vi) WST d WST,

(vii) WST 1

d WST1

if d, d d .

Proposition 1.3.2.2.[24].Suppose f : a, b A is internal,-continuous, and monotonic. Then(1) fhas a unique continuous extension f# a, b d A

d, where

A denotes the closure ofA in d.(2)The conclusion (1) above holds iff is piecewise monotonic

(i.e., the domain can be decomposed into a finite (not -finite) numberof intervals on each of which fis monotonic).Proposition 1.3.2.3.[24].Suppose f,gare -continuous, piecewisemonotonic functions then

(i) f g is also and

(ii) f g# f# g# .

I.3.3.Absorption numbers in d and idempotents.

I.3.3.1.Absorption function and numbers in d.One of standard ways of defining the completion of involves restricting

oneself to subsets a which have the following property 0xx yy y x .

It is well known that in this case we obtain a field. In fact the proof is essentially the

same as the one used in the case of ordinary Dedekind cuts in the development of

the standard real numbers, d, of course, does not have the above property

because no infinitesimal works.This suggests the introduction of the concept ofabsorption part ab. p. of a number for an element of d which, roughly


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speaking, measures how much a departs from having the above property [23]. We

also introduce similar concept of an absorption number ab. n. ab. n. , (cut) for given element of d.

Definition 1.3.3.1.1.[23].ab. p. d 0|xxx d .

Example 1.3.3.1.(i) : ab. p. 0,(ii) ab. p. d d, (iii) ab. p. d d,(iv) : ab. p. d d,(v) : ab. p. d d.Definition 1.3.3.2. ab. n. , .

Example 1.3.3.2.(i) 0 : ab. n. d, ,(ii) ab. n. d, d , ab. n. d, d , ab. n. d, d ,(iii) : ab. n. d, d , ab. n. d, d , ab. n. d, d ,(iv) : ab. n. d, ,

(v) ab. n. d, d , ab. n. d, d , ab. n. d, d .

Lemma 1.3.3.1.[23].(i) c ab. p. and 0 d c d ab. p. (ii) c ab. p. and d ab. p. c d ab. p. .Remark 1.3.3.1. By Lemma 1.3.2.1 ab. p. may be regarded as an

element of d by adding on all negative elements ofd to ab. p. .

Of course if the condition d 0 in the definition of ab. p. is deleted weautomatically get all the negative elements to be in ab. p. since

x y x . The reason for our definition is that the real interest liesin the non-negative numbers. A technicality occurs if

ab.

p.

0. We

then identify ab. p. with 0. [ab. p. becomes x|x 0 which by ourearly convention is not in d].

Remark 1.3.3.1.2.By Lemma 1.3.2.1(ii), ab. p. is idempotent.

Lemma 1.3.3.1.2.[23].

(i) ab. p. is the maximum element d such that .(ii) ab. p. for 0.(iii) If is positive and idempotent then ab. p. .

Lemma 1.3.3.1.3.[23]. Let d satsify 0. Then the following are

equivalent. In what follows assume a, b 0.(i) is idempotent,

(ii) a, b a b ,(iii) a 2a ,(iv) nna n a ,(v) a r a , for all finite r .

Connection with the value group.Definition 1.3.3.1.2. We define an equivalence relation on the positive


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elements of as follows: a ~ b ab

and ba are finite.Then the equivalence

classes from a linear ordered set. We denote the order relation by .The classes may be regarded as orders of infinity.

The subring of consisting of the finite elements is a valuation ring, and theequivalence classes may also be regarded as elements of the value group.

Condition (v) in Lemma 1.3.3.1.3 essentially says that a and b~a b ,i.e. a may be regarded as a Dedekind cut in the value group.

Properties of the Absorption Function.

Theorem 1.3.3.1.1.[23]. ab. p. .

Theorem 1.3.3.1.2.[23].ab. p. ab. p. .

Theorem 1.3.3.1.3.[23].

(i) ab. p. .(ii) ab. p. .

Theorem 1.3.3.1.4.[23].

(i) ab. p. ab. p. ,

(ii) ab. p. maxab. p. , ab. p. .

We now classify the elements such that . For positive we

know by Lemma 1.3.3.1.2.(i) that iff ab. p. .Theorem 1.3.3.1.5.[23]. Assume 0. If absorbs then a abosrbs .

Theorem 1.3.3.1.6.[23]. Let 0 d. Then the following are equivalent

(i) is an idempotent,

(ii) ,(iii) .

Special Equivalence Relations on d.Let be a positive idempotent. We define three equivalence relations

R , S and T on d.

Definition 1.3.3.1.3.[23].


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(i) Rmod ,

(ii) Smod ,(iii) Tmod dd d d.

Remark 1.3.3.1.3.To simplify the notation mod is omitted when we are

dealing with only one . R and S are obviously equivalence relations.

T is an equivalence relation since is idempotent.

Remark 1.3.3.1.4.It is immediate that R, S and T are congruence relations

with respect to addition. Also, if ~ stands for either R, S orT then

and ~ ~ . To see this it is convenient to have the following lemma.

Lemma 1.3.3.1.4.[23]. Suppose . Then

(i) Rmod

,(ii) Smod .

Lemma 1.3.3.1.5.[23]. Let be a positive idempotent. Then

.Remark 1.3.3.1.5.This is not immediate since the inequality

goes the wrong way. In fact, this seems surprising atfirst since the first addend may be bigger than one intuitively expects, e.g. if

d then d d d 0. However,d d d, so the inequality is valid after all.

Theorem 1.3.3.1.7.[23].

(i) S is a congruence relation with respect to negation.

(ii) T is a congruence relation with respect to negation.

(iii) R is not a congruence relation with respect to negation.

Theorem 1.3.3.1.8.[23]. is the maximum element satisfying R.

Theorem 1.3.3.1.9.[23]. is the minimum element satisfying S.Theorem 1.3.3.1.9.[23]. T R S. Both inclusions are proper.

Theorem 1.3.3.1.10.[23].

(i) Let 1 and 2 be two positive idempotents such that 2 1. Then:

2 1 2,(ii) Let 1 and 2 be two positive idempotents such that 2 1. Then:

Smod 1 Rmod 2 .

Theorem 1.3.3.1.11.[23].Let 1 and 2 be two positive idempotents such

that 2 1. Then Smod 1 Tmod 2 but not conversely.

Theorem 1.3.3.1.12.[23].S is the smallest congruence relation with respectto addition and negation containing R.


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Theorem 1.3.3.1.13.[23].Any convex congruence relation ~ containing T

properly must contain S.

I.3.3.2.Special Kinds of Idempotents in d

.

Let a such that a 0. Then a gives rise to two idempotents in anatural way.

Definition 1.3.3.2.1.[23].

(i) Aa x|nnx n a.(ii) Ba x|rr x r a.Then it is immediate that Aa and Ba are idempotents.The usual "/2 argument"

shows this for Ba. It is also clear that Aa is the smallest idempotent containing a

and Ba is the largest idempotent not containing a. It follows that Ba and Aa areconsecutive idempotents.

Remark 1.3.3.2.1.Note that B1 d inf (which is the set of all infinite

small positive numbers plus all negative numbers) which we have already

considered above. A1 d sup (which is the set of all finite numbers

plus all negative numbers) which we have also already considered above.

Definition 1.3.3.2.2. Let a .

(i) da x|nnx n a.(ii) da x|rr x r a, a

Remark 1.3.3.2.2.Then it is immediate that da and da are idempotents.It is also clear that da is the smallest idempotent containing hypernatural aand da a d. da a d is the largest idempotent not containing a.It follows that da and da are consecutive idempotents.Remark 1.3.3.2.3. Note that d1 d (which is the set of all finite natural

numbers plus all negative numbers) which we have also already consideredabove.

Theorem 1.3.3.2.1.[23].

(i) No idempotent of the form Aa has an immediate successor.

(ii) All consecutive pairs of idempotents have the form Aa and Bafor some a .


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I.3.3.3. Types of with a given ab. p. .

Among elements of d such that ab. p. we can distinguishtwo types.

Definition 1.3.3.3.1.[23]. Assume 0.

(i) d has type 1 ifxx yx y y ,(ii) d has type 2 ifxx yy x y , i.e. d has type 2 iff does not have type 1.A similar classification exists from above.

Definition 1.3.3.3.2.[23]. Assume 0.

(i) d has type 1A ifxx yx y y ,

(ii) d has type 2A ifxx yy x y .

Theorem 1.3.3.3.3.[23].

(i) d has type 1 iff has type 1A,(ii) d cannot have type 1 and type 1A simultaneously.

Theorem 1.3.3.3.4.[23].Suppose ab. p. 0. Then has type 1iff has the form a for some a .Theorem 1.3.3.3.5.[23]. d has type 1A iff has the form a for some a .Theorem 1.3.3.3.6.[23].

(i) Ifab. p. ab. p. then has type 1 iff has type 1.

(ii) Ifab. p. ab. p. then has type 2 iff either or

has type 2.

Theorem 1.3.3.3.7.[23]. Ifab. p. has the form Ba then has

type 1 or type 1A.

I.3.3.4. -Part of with ab. p. 0.Theorem 1. 3. 3. 4. 8. (i) Suppose:

1) d d,2) ab. p. d i.e. has type 1.


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Then there is exist unique a such that

a# d,

a WST.

1.3.3.5

(ii) Suppose:

1) d 1 d, d 2 d,2) ab. p. 1 d, ab. p. 2 d i.e. 1 and 2 has type 1.

Then:

1 2 WST1 WST2 d. 1.3.3.6

(iii) Suppose:

1) d d,2) ab. p. d i.e. has type 1.

Then b b

:

b#

b#

WST#

b#

d. 1.3.3.7

.

(iv) Suppose:

1) d 1 d, d 2 d,2) ab. p. 1 d, ab. p. 2 d i.e. 1 and 2 has type 1.

Then b b

:

b# 1 2 b# WST#

b# WST2 #

b# d. 1.3.3.8

Theorem 1. 3. 3. 4. 9. (i) Suppose:

1) d d,2) ab. p. d i.e. has type 1A.Then there is exist unique a such that


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a# d.

a WST.

1.3.3.9

(ii) Suppose:

1) d 1 d, d 2 d,2) ab. p. 1 d, ab. p. 2 d i.e. 1 and 2 has type 1A or3) ab. p. 1 d, ab. p. 2 d i.e. 1 has type 1 and 2 hastype 1A. Then:

1 2 WST1 WST2 d. 1.3.3.10

(iii) Suppose:

1) d d,2) ab. p. d i.e. has type 1A.Then b b

:

b# b# WST# b# d. 1.3.3.11

.

(iv) Suppose:

1) d 1 d, d 2 d,2) ab. p. 1 d, ab. p. 2 d i.e. 1 and 2 has type 1A or3) ab. p. 1 d, ab. p. 2 d i.e. 1 has type 1 and 2 hastype 1A. Then b b

:

b# 1 2 b# WST# b# WST2 # b# d. 1.3.3.12

Definition 1. 3. 3. 4. 3.Suppose ab. p. d i.e. has type 1,

i.e. a# d, a WST, a .Then

, ( 0, ) is an -part of iff:


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y a# y a# y y # d . 1.3.3.13

Theorem 1. 3. 3. 4. 10. Suppose d d, ab. p. di.e. has type 1, d b d, b , c .Then

(i)

a# # d.

(ii) b#

stb # a# # d.

(iii) c#

c# a# # d.

Theorem 1. 3. 3. 4. 11. (i) Suppose d 1 d, d 2 d,

ab. p. 1 ab. p. 2 d, WST1 a , WST2 b .Then 1 2

a# b# # d.

(ii) 1 2 a# b# # d.

Theorem 1. 3. 3. 4. 11. 0 d # d .

Definition 1. 3. 3. 4. 4. (i) Suppose ab. p. d, and hastype 1, i.e. has representation a# for some a , a# .Then |a#

, ( 0, ) is an -part of for a given a iff:

ya# y a# y y # . 1.3.3.14

Suppose ab. p. d, . Then |,

0, .is an -part of for a given a iff:

yy y y # . 1.3.3.15

Note ifab. p. d and a#

d, a then |a#

,

is an -part of.

Definition 1.3.3.4.4. Suppose ab. p. d and has type 1A, i.e.i.e. has representation a# for somea , a# . Then |a#

is an -part of for a given

a iff:


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ya# y a# y y # . 1.3.3.16

Note ifab. p. d i.e. a# d, a then |a#

,

is an -part of.

Theorem 1.3.3.3.10.

(1) Suppose ab. p. d and has type 1,i.e. a# for some a .Then |a#

has the form

|a#

a# # 1.3.3.17

for a given a .(2) Suppose ab. p. d and has type 1A,i.e. a# for some a .Then |a#

has the form

|a#

a# # 1.3.3.18

for a given a .

Theorem 1. 3. 3. 4. 11.

(1) Suppose ab. p. d i.e. has type 1 and

has representation a# d, for some unique a .Then

has the unique form:

a# # d. 1.3.3.19

(2) Suppose ab. p. d i.e. has type 1A and

has representation a# d, for some unique a .


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Then

has the unique form:

a# d. 1.3.3.20

Theorem 1. 3. 3. 4. 12. (1) Suppose ab. p. d, WST 0i.e. has type 1 and has representation a# d,for some unique a . Then for every M

M

M M a#

M a# # M d.

1.3.3.21

(2) Suppose ab. p. d, WST 0and has type 1 i.e. a# d, for some unique a .Then for every M

M

M M a#

M a# # M d.

1.3.3.22

Theorem 1.3.3.4.13. (i) Suppose ab. p. d i.e. has type 1.

Then d y 0y y y # d .

(ii) Suppose ab. p. d i.e. has type 1A.Then d y 0y y y

# d .

I.3.3.5.Multiplicative idempotents.

Definition 1.3.3.5.1.[23]. Let Sd x|yy Sx y.Then Sd satisfiesthe usual axioms for a closure operation.

Let fbe a continuous strictly increasing function in each variable from asubset ofn into . Specifically, we want the domain to be the cartesian


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product i1n A i, where A i x|x a i for some a i . By transfer fextends

to a function ffrom the corresponding subset of n into which is alsostrictly increasing in each variable and continuous in the Q topology (i.e. and range over arbitrary positive elements in ).

Definition 1.3.3.5.2.[23]. Let i d, bi , then

fd1, 2, . . . , n fb1, b2, . . . , bn | b i i

d

Theorem 1.3.3.5.1.[23]. Iffand gare functions of one variable then

f gd fd gd.

Theorem 1.3.3.5.2.[23].Let fand gbe any two terms obtained bycompositions of strictly increasing continuous functions possibly containing

parameters in . Then any relation f gor f gvalid in extends tod, i.e. fd gd orfd gd.

Theorem 1.3.3.5.3.[23].The map expd maps the set of additive

idempotents onto the set of all multiplicative idempotents other than 0.

Similarly, multiplicative absorption can be defined and reduced to the study

of additive absorption. Incidentally the map expd is essentially the

same as the map in [34, Theorem 6] which is the map from the set of ideals

onto the set of all prime ideals of the valuation ring consisting of the finite

elements of .

I.3.3.6.Additive monoid of Dedekind hyperreal

integers d.

Well-order relation s (or strong well-ordering) on a set S is a total orderon Swith the property: that every non-empty subset S ofShas a least elementin this ordering.The set Stogether with the well-order relation s is thencalled a (strong) well-ordered set.

The natural numbers of with the well-order relation are not strongwell-ordered set,for there is no smallest infinite one.


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Definition 1.3.3.6.1. Weak well-order relation w (or weak well-ordering)

on

a set Sis a total order on Swith the property:every non-empty subset S Shasa least element in this ordering or S has a greatest lower bound (infS ) in this

ordering.The set Stogether with the weak well-order relation w is then called aweak well-ordered set.The natural numbers of with the well-order relation are not ivenweak well-ordered set,for there is no infS in S.Let us considered completion of the ring .Possible standard completion

of the ring can be constructed by Dedekind sections. Making a semantic

leap, we now answer the question:"what is a Dedekind hyperintegers ?"

Definition 1.3.3.6.2. A Dedekind hyperinteger is a cut in . d d, is the class of all Dedekind hyperintegers x A|B,A ,B x A,A .We will show that in a natural way d is a complete ordered additive

monoid containing .

Before spelling out what this means, here are some examples of cuts in .

(i)

A|B n

| n 1 n

| n 1 .

(ii)

A|B n

| n n

| n , where .

(iii)

A|B n

|n 0 n n

| n

.

(iv)

A|B

n

|n 0 n

i

n i

n

| n

i

n i ,

where .Remark. 1.3.3.6.1. It is convenient to say that A|B d is an integer(hyperinteger) cut in if it is like the cut in examples (i),(ii): fore some

fixed integer (hyperinteger) number c ,A is the set of all integern such that n c while B is the rest of .The B-set of an integer (hyperinteger) cut contains a smollest c, andconversaly ifA|B is a cut in and B contains a smollest element c thenA|B is an integer (hyperinteger) cut at c. We write c for the integer andhyperinteger cut at c. This lets us think of d by identifying c with c.Remark.1.3.3.6.2. It is convenient to say that:

(1) A|B d is an standard cut in if it is like the cut in example (i):

fore some cut A

|B

the next equality is satisfied:A|B A

|B

,i.e. A-set of a cut is an standard set.


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(2) A|B d is an internal cut ornonstandard cut in if it is like the cutin example (ii), i.e. A-set of a cut is an internal nonstandard set.(3) A|B d is an external cut in if it is like the cut in examples (iii)-(iv),i.e. A-set of a cut is an external set.

Definition 1.3.3.6.3. A Dedekind cut in

is a subset

of thehyperinteger numbers that satisfies these properties:

1. is not empty.

2. \ is not empty.

3. contains no greatest element.

4. Forx,y , ifx and y x, then y as well.Definition 1.3.3.6.4. A Dedekind hyperinteger d is aDedekind cut in . We denote the set of all Dedekind hyperinteger

by d and we order them by set-theoretic inclusion, that is to

say, for any , d, d (or ) if and only if where the

inclusion is strict. We further define d (or ) as hyperinteger

if and are equal as sets. As usual, we write d if d

or d .

Definition 1.3.3.6.5. M d is an upper bound for a set S d if eachs Ssatisfies s d M. We also say that the set Sis bounded above byMiffM L. We also say that the set Sis hyperbounded above iffM L, i.e. |M| \.Definition 1.3.3.6.6. An upper bound for Sthat is less than all other upperbound forSis a least upper bound forS.

Theorem 1.3.3.6.1 Every nonempty subset A d of Dedekindhyperinteger that is bounded (hyperbounded) above has a least

upper bound.

Definition 1.3.3.6.7. Given two Dedekind hyperinteger and we

define:

1.The additive identity (zero cut) denoted 0 d or0, is

0 d x

| x 0 .

2.The multiplicative identity denoted 1 d or1, is

1 d x

| x 1 .

3. Addition d of and also denoted is

d x y| x ,y .

It is easy to see that d 0 d 0 d for all d.

It is easy to see that d is a cut in and d d .

Another fundamental property of cut addition is associativity:

d d d d .

This follows from the corresponding property of .

4.The opposite d of, also denoted , is

x

| x , x is not the least element of

\ .

5.Remark 1.3.3.6.3. We also say that the opposite of is the additive


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inverse of denoted iff the next equality is satisfied: 0.

6.Remark 1.3.3.6.4. It is easy to see that for all standard and internal cut Int

the

opposite Int is the additive inverse ofInt, i.e. Int Int 0.

7.We say that the cut is positive if0 or negative if 0.The absolute value of , denoted ||, is || , if 0 and || , if 0.8. The cut order enjois on d the standard additional properties of:

(i) transitivity: d d d .

(ii) trichotomy: eizer d , d or d but only one of the

three things is true.

(iii) translation: d d d d .

9.By definition above, this is what we mean when we say that d is an complete ordered additive monoid.

Remark 1.3.3.6.5. Let us considered Dedekind integer cut c

d as subset ofc d. We write

c d-supc x

sup x|x c

d for the cut c d.

This lets us think of canonical imbeding djd

d monoid

d into generalized pseudo-field d

d

d

by identifying c with it image

c jdc.Remark 1.3.3.6.6. It is convenient to identify monoid d with it

image jd d d.

I.3.5.Pseudo-ring of Wattenberg hyperreal

integers d.

The set d has within it a setd

d of Wattenberg hyperreal integers

which behave very much like hyperreals inside . In particular the

greatest integer function : extends in a natural way

to d :d

d.

Lemma 1.3.5.1.[24].Suppose d. Then the following twoconditions on are equivalent:

(i) sup #|

d ,

(ii) inf #|

d .

Definition 1.3.5.7.If satisfies conditions (i) or (ii) from lemma 1.3.5.1

is said to be a d

-integer or Wattenberg (hyperreal) integer.

Lemma 1.3.5.2.[24].(i) d is the closure ind of

,


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(ii)

d is the closure ind of

,

(iii) both d and d are closed with respect to taking sup

and inf.

(iv)

d is a weak well-ordered set.

Lemma 1.3.5.3.[24].Suppose that ,

d. Then,(i) d

d.

(ii) d d.

(iii) d d.

Definition 1.3.5.8. Suppose d. Then, we define d :

d d by: d sup |

d .

Remark 1.3.5.7.There are two possibilities:

(i) collection |

d has no greatest element. In this

case d

since d

d implies a a

d

d a

d .But then a d d which implies a d d 1 d d contradictingwith d

d a d a d d 1 d(ii) collection |

d has a greatest element,. In this

case d .

Definition 1.3.5.9. d-integersup inf we denote d.

Definition 1.3.5.10. Suppose . Then, we define -block bkas a set of hyper integers such that bk n|n .For, there are two possibilities:

(i) . In this case bk bk and we write bk bk where

/.(ii) | | . In this case bk bk and bk

bk

.

Lemma 1.3.5.11. bk .

Proof. Clear by using [25,Chapt.1,section 9].

I.3.6.External summation of countable and hyperfinite

sequences in

d.Definition 1.3.6.1. Let SX denote the group of permutations of the set Xand HX

denote ultrafilter on the set X. Permutation SX is admissible iff preservHX, i.e. for any A HX the next condition is satisfied: A HX.

Below we denote by SX,HX the subgroup SX,HX SX of the all admissible

permutations.

Definition 1.3.6.2. Let us consider countable sequence sn : , such that

(a) nsn 0 or (b) nsn 0 and hyperreal number denoted sn whichformed from sequence sn n by the law


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sn s0, s0 s1, s0 s1 s2, . . . ,0i

si, . . .

. 1.3.6.1

Then external sum of the countable sequence sn denoted

Ext- n

sn 1.3.6.2

is

a : Ext n

sn inf sn | S,H ,

b : Ext n

sn sup sn | S,H

1.3.6.3

accordingly.

Example 1.3.6.1. Let us consider countable sequence 1n n such

that: n1n 1. Hence 1n 1,2,3,. . . . , i, . . . and usingEq.(1.3.3) one obtain

Ext n

1n

. 1.3.6.4

Example 1.3.6.2. Let us consider countable sequence 1n

n such that:

n|1n

1 H. Hence 1n 1,2,3,. . . . , i, . . . modH and

using Eq.(1.3.3) one obtain

Ext n

1n

. 1.3.6.5

Example 1.3.6.3. (Eulers infinite number E#). Let us consider countablesequence hn n1. Hence


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hn 1, 1 12

, 1 12

13

, . . . . , 1 12

13

. . . 1i

, . . .

and using Eq.(1.3.3) one obtain

Ext n1

hn E# d. 1.3.6.6

Definition 1.3.6.8. Let us consider countable sequence sn : and

two subsequences denoted sn : , sn

: which formed from

sequence sn n by the law

sn sn sn 0,

sn 0 sn 0

1.3.6.7

and accordingly by the law

sn sn sn 0,

sn 0 sn 0

1.3.6.8

Hence sn n sn sn

n.

Example 1.3.6.4. Let us consider countable sequence

1n

n 1, 1,1, 1,.. . ,1, 1 , . . . . 1.3.6.9

Hence 1n

n 1n

1n

n where


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1n n 1,0,1,0,. . . ,1,0,. . .

1n n 0, 1,0, 1,.. . ,0, 1 , . . . .

1.3.6.10

Definition 1.3.6.9.The external sum of the arbitrary countable

sequence sn n denoted

Ext n

sn 1.3.6.12

is

Ext n

sn Ext n

sn Ext

n

sn . 1.3.6.13

Example 1.3.6.5. Let us consider countable sequence (1.3.9) Using

Eq.(1.3.3),Eq.(1.3.13) and Eq.(1.3.5) one obtain

Ext n

1nn

Ext n

1n

Ext n

1n

0.

1.3.14

Definition 1.3.6.10. Let us consider countable sequence sn#

:

d,such that

(a) nsn# 0 or (b) nsn# 0.Then external sum of the countable sequence sn

# denoted

#Ext- n

sn#

1.3.6.15

is


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a : #Ext n

sn#

k

sup nk

sn# S ,

b : #Ext n

sn#

k

inf nk

sn# | S .

1.3.6.16

Definition 1.3.6.11. Let us consider countable sequence sn# : c

and two subsequences denoted #sn : c,

#sn : c which

formed from sequence sn# n by the law

#sn sn sn

# 0,

#sn 0 sn

# 0

1.3.6.17

and accordingly by the law

#sn sn

# sn# 0,

#sn 0 sn

# 0

1.3.6.18

Hence sn# n

#sn

#sn

n.

Definition 1.3.6.12.The external sum of the arbitrary countable

sequence sn n denoted

#Ext n

sn#

1.3.6.19

is

#Ext n

sn# #Ext

n

#sn #Ext

n

#sn . 1.3.6.20


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Definition 1.3.6.13. Let us consider an nonempty subset A d which

is bounded or hyperbounded from above and such that:

supA supA for any 0. We call this least upper bound supAthe strong least upper bound orstrong supremum, written as s-supA.

Proposition 1.3.6.1. IfA is a nonempty subset of d which is bounded

from above and strong supremum s-supA exist, then:

(1) s-supA is the unique number such that s-supA is an upper bound

forA and s-supA is not a upper bound for A for any 0, 0;(2) (The Strong Approximation Property) let 0, 0 there exist x Asuch that s-supA x s-supA.Proof.(2) If not, then s-supA is an upper bound ofA less than the leastupper bound, which is a contradiction.

Corollary 1.3.6.1. Let A be bounded or hyperbounded from above andnon-empty set such that s-supA exist. There is a function

: d such that for all n we haves-supA n1 n s-supATheorem 1.3.6.2. Let A be a non-empty set which is bounded or

hyperbounded from below. Then the set of lower bounds of A has a

greatest element.

Proof. Let A x|x A. We know that (i) xx d yy d x y y x.

Let lA be a lower bound ofA.Then lA x for all x A. So x lA for all

x A, that is y lA for all y A. So A is bounded above, and non-empty,so by the Theorem 1.3.1 supA exists.We shall prove now that: (ii) supA is a lower bound ofA, (iii) iflA is a lowerbound ofA then lA supA. (ii) ifx A then x A and so x supAHence by statement (i) x supA and we see that supA is a lowerbound ofA. (iii) If lA x for all x A then lA y for all y A.HencelA supA by virtue ofsupA being the least upper bound of A.Finally we obtain: lA supA.Definition 1.3.6.14. We call this greatest element a greatest lower bound

or infinum ofA,written is infA.

Definition 1.3.6.15.Let us consider an nonempty subset A d which

is bounded or hyperbounded from below and such that:

infA infA for any 0, 0.We call this greatest lower bound infA a strong greatest lower boundor

strong infinum, written is s-infA.

Definition 1.3.6.16.Let us consider an nonempty subset A d which


(1) there exist 0 0 such that infA 0 infA,(2) infA infA for any 0 such that 0 0.

We call this greatest lower bound infA almost strong greatest lower boundoralmost strong infinum, written is os-infA.


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Definition 1.3.6.17. Let us consider an nonempty subset A d which


(1) infA infA for any 0, 0,(2) infA infA for any 0 such that 0.

We call this greatest lower bound infA weak greatest lower boundorweak infinum, written is w-infA.Definition 1.3.6.18. Let us consider an nonempty subset A d which

is hyperbounded from below and such that:

(1) infA infA for any 0, ,(2) infA infA for any 0 such that .We call this greatest lower bound infA ultra weak greatest lower bound

orultra weak infinum, written is uw-infA.Proposition 1.3.6.2. (1) IfA is a nonempty subset of d which is bounded

from below and strong infinum s-infA exist, then:

s-infA is the unique number such that s-infA is an upper bound

forA and s-infA is not a lower bound for A for any 0, 0;(2) IfA is a nonempty subset of d which is bounded from above, then:

s-supA is the unique number such that s-supA is an upper bound for A

and s-supA is not an upper bound forA for any 0, 0.Proposition 1.3.6.3.(a). (Strong Approximation Property.)

(1) IfA is a nonempty subset of d which is bounded (hyperbounded)

from above and such that strong supremum s-supA exist, and let

0, 0 there exist x A such that s-supA x s-supA.

(2) IfA is a nonempty subset of d which is bounded (hyperbounded)from below and such that strong infinum s-infA exist, and let

0, 0 there exist x A such that s-infA x s-infA .Proof. (1) If not, then s-supA is an upper bound ofA less than thestrong upper bound s-supA, which is a contradiction.

(2) If not, then s-infA is an lower bound of A bigger than the

strong lower bound s-infA, which is a contradiction.

Proposition 1.3.6.3.(b) (The Almost Strong Approximation Property.)

(1) IfA is a nonempty subset of d which

Non-Archimedean Analysis on the Extended Hyperreal Line *R_d and some transcendence conjectures over field field *Q and *Q_omega .

Documents

Non-Archimedean Analysis on the Extended Hyperreal Line R_d and some transcendence conjectures over field field Q and *Q_omega .