The new Fair Natural Mathematics without the infinite, of the 2nd Millennium. : 6. The Axiomatic multi-precision decimal Digital or Democritus Real Numbers.

THIS PROJECT IS UNDER THE NEXT PHILOSOPHICAL PRINCIPLES

1) CONSCIOUSNESS IS INFINITE. CONVERSELY, THE INFINITE IS A FUNCTION AND PROPERTY OF THE CONSCIOUSNESSES.

2) BUT THE PHYSICAL MATERIAL WORLD IS FINITE.

3) THEREFORE MATHEMATICAL MODELS IN THEIR ONTOLOGY SHOULD CONTAIN ONLY FINITE ENTITIES AND SHOULD NOT INVOLVE THE INFINITE.

THIS PROJECT THEREFORE IS CREATING AGAIN THE BASIC OF MATHEMATICS AND ITS ONTOLOGY WITH NEW AXIOMS THAT DO NOT INVOLVE THE INFINITE AT ALL.

The Digital Real Line, the Digital Fractional Real Numbers and the Digital Decimal Real Numbers
Our perception and experience of the reality depends on the system of beliefs that we have. In mathematics, the system of spiritual beliefs is nothing else than the axioms of the axiomatic systems that we accept. The rest is the work of reasoning and acting.

The abstraction of the infinite seems sweet at the beginning as it reduces some complexity, in the definitions, but later on it turns out to be bitter, as it traps the mathematical minds in to a vast complexity irrelevant to real life applications.

Still another way to understand the digital or natural mathematics is to realize that they are essential a detailed and systematic presentation of what actually the scientists that apply the classical mathematics (meteorologist, physicists, biologist, ecologist, economists, etc.) are forced to do since they cannot apply neither infinite steps, or objects of infinite information, neither pretend that they have infinite many material atoms. Thus what they do is logically not equivalent to what the classical mathematics are , Sometimes they call it approximation, but essentially it deserves a totally new terminology and presentation. This adds to the applied mathematical scientists a layer of better consciousness and clearer awareness of what they are doing in mathematical terms. So that what they say, what they do and what they see has congruence greater sincerity and integrity

Before the discussion below for creation of an axiomatic system of the axiomatic digital decimal real numbers , and axiomatic digital fractional real numbers, we state here a third primary approach for the axiomatic digital real numbers through the axioms of (axiomatic digital) euclidean straight line (see post 7). This approach goes back to Euclid and Eudoxus (see e.g. http://en.wikipedia.org/wiki/Eudoxus_of_Cnidus) who defined number magnitudes as ratios of euclidean straight line segments. The initial concepts here are not the operations of addition and multiplication, but order, incidence, equality or congruence and a new initial concept not existing in ancient Greek mathematics, that of (finite) resolution.

PEANO-LIKE AXIOMS, for the digital natural numbers

We define the natural numbers in two scales (and later precision levels) that are two unequal initial segments of the natural numbers N(ω)< N(Ω) .The number ω is called the Ordinal size ω of the local system of natural numbers Ν(ω) while the Ω is the cardinal size of the global system of natural numbers. ω<Ω. If we start with integers n1, n2, n3 from N(ω), then their addition and multiplication, have the commutative semiring properties but without closure in N(ω), but with values in N(Ω). We call the N(ω), the local segment while the N(Ω) the global segment.
We have here an initial relation among the natural numbers which is called successor or next of a natural number x and it is denoted by S(x).

1) The number 1 is a natural number and belongs both to N(ω), and N(Ω) .

2) There is no natural number whose successor is 1.

3) If x is a natural number of N(ω), its successor S(x) , is also a natural number belonging in N(Ω).

4) If two different numbers of N(Ω) , have the same successor, then they are equal, Formally if S(x)=S(y) then x=y .

5) (Peano axiom of induction) If a property or formal proposition P() holds for 1 (that is P(1)=true) and if when holding for x in N(ω) holds also for P(S(x)) with S(x) in N(Ω) , then it holds for all natural numbers of N(ω).

6) Axiom of sufficient large size. If we repeat the operations of the commutative semiring starting from elements of the local version N(ω), ω-times, the results are still inside the larger set N(Ω).

This last Peano axiom of induction is useful only if the natural numbers are formulated within a formal logic (the axiom itself as a formal proposition is in 2nd order formal logic) that its size Ω(l) is less than the size of the objective system of natural numbers Ω. Otherwise for sufficient large

Ω(L)>>Ω, we may simply construct a lengthy proof of this axiom starting from P(1) then P(2) ...and finally P(Ω), which then it is a theorem.

Any two models Μ1 Μ2 of the digital natural numbers Ν(Ω),Ν(ω) of equal size ω, Ω are isomorphic.

AXIOMATIC SYSTEM OF THE (DIGITAL) REAL LINE

Creating 3D mental images with the imagination

The physiscists andd nobel prize winner R Feynamnn often was explaining that he was training his imagination imagining in 3D mental images the moving molecules of a gas in box, and perceiving that temperature as their avearge velocity, the ressure as the average moemntum of the walls of the box, and the density as the average number of particles per unit of volume, He was explaing that he was getting much pleasure from this spiritual excerise of craeting 3D mental images and simulating the reality. Excatly this 3D VISUAL MENTAL IMAGES LANGUAGE parctice is UTILIZED BY ME AND is required by someone to understand and develop the digital or natural mathematics, in geometry, real numbers and analysis.

Rules for phantasy and drawing of figures.

As initially we considered a system of digital real numbers R(m,n,p,q) we consider the points of P(m), P(n) as visible in the figures while the points of P(p) as invisble pixels , and those of P(q) as invisible atoms. Therefore, even the points and seemingly infinitesimals that will be defined below, of P(n) relative to P(m) are considered as visible. This is in accordance with the habit in classical mathematics to make the points visible, although they claim that they have zero size.

We have as initial concept of objects

a) The High resolution or precision points , or invisible points or pixels or atoms

b) The Low resolution or precision points, or visible points. They have the minimum no-zero size (length, width or diameter ) in the Low resolution real numbers that can be created on a geometric line as we shall see.

The Low precision or visible congruence of visible linear segments AB of two visible points points A, B,

c) The visible points

b) We may apply the digital set theory on the points of the digital real numbers

d) And of course we may apply the digital formal logic to make arguments and proofs.

e) Besides the congruence equivalence relations we have the next initial relations

Among visible or invisible elements.

An invisible point A belongs to a visible point B , denoted by A ε B

A visible point A belongs to the real line L or , denoted by A ε L

A visible point A is between two visible points B, C.

We design 5 groups of axioms

1) Of Incidence
2) Of Order
3) Of Congruence
4) Of Continuity
5) Of Resolution

In the next axioms the term point refers to visible or low precision point. It has the minimum no-zero size (length) in the Low resolution real numbers that can be constructed on a real line as we shall see. may be created on a real line. We use the axioms of Hilbert, but we modify them and add more axioms.
I. Incidence

There exist at least two points on the real line.
There is a special point denoted by O, which is called the center or origin of the real line, and a special linear segment OA, which is called the unit of measurement of lengths in the real line.
For every invisible point A, there is a visible point B, so that A belongs to B.
Two invisible points A, B belong to the same visible point C is an equivalence relation among the invisible points.
All points (visible or invisible) belong to the real line.

II. Order

If a point B lies between points A and C, B is also between C and A.
Of any three points situated on the real line, there is no more than one which lies between the other two.
The real line , has two points ω1 and ω2 so that every other point of the line , lies between ω1 and ω2 . We call them the end points of the real line.

III. Congruence

If A, B are two points on the real line a, and if A′ is a point upon the same , then, upon a given side of A′ on the straight line a , and as long as the the A'ω2 is larger than the AB, we can always find a point B′ so that the segment AB is congruent to the segment A′B′ . We indicate this relation by writing AB ≅ A′ B′. Every segment is congruent to itself; that is, we always have AB ≅AB.
We can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of the real line in at least one way.
If a segment AB is congruent to the segment A′B′ and also to the segment A″B″, then the segment A′B′ is congruent to the segment A″B″; that is, if AB ≅ A′B′ and AB ≅ A″B″, then A′B′ ≅ A″B″.
Let AB and BC be two segments of the real line a which have no points in common aside from the point B, and, furthermore, let A′B′ and B′C′ be two segments of the real line a having, likewise, no point other than B′ in common. Then, if AB ≅ A′B′ and BC ≅ B′C′, we have AC ≅ A′C′.

V. Continuity

Axiom of Archimedes. If AB and CD are any segments of he real line , with AB<CD , then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B, as long as Aω2 is at least larger than twice the AB.
Axiom of line completeness. An extension of a set of visible points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III and from V-1 is impossible. (this axioms turns out to be redundant but for historic reasons we state it here. It defines essentially what visible is relative to invisible. And that the distinction of visible and invisible cannot be refined in the sense that any smaller size than the visible points is invisible)

VI. Resolution (or Density)

Axiom of sufficient high resolution or density . Let the real line with its origin O and the units of measurements OA on it, and let ω(a), Ω(a), demote the finite cardinal number of visible and invisible points that belong to the real line . And let ω(n) be the size of the model of the natural numbers constructed on the real line through congruence and the above axioms. Then it holds that

2^ω(n) <= ω(a)
2^ω(a) <=Ω(a)

Let us denote by ω(R) the finite cardinal number of all visible points, and by Ω(R) the finite cardinal number of all invisible points. Let the real line with its origin O, and let us denote by ω(n) is the size of the model of the natural numbers constructed on the real line and the unit of measurements OA, through congruence and the above axioms.

If two set-theoretical models M1, M2 of the above real line have the same , ω(n), ω(R),Ω(R) , then they are isomporphic under appropriate conditions and restrictions.

In other words , the above axiomatic system is in some particular sense (not absolute) categorical up to the units of measurements and visible and invisible density. To achieve absolute categorical axiomatic system, some assumptions about how many invisible points has each visible point , if all visible points have equal number of invisible points etc must be made.

Further more there is an alternative axiomatic system, in which we have endowed the invisible points or material atoms and invisible atomic lines and planes, with a similar geometric structure , as the visible points, lines and planes. The geometric structure of the invisible atomic points, is again, that of incidence, order, congruence and resolution. We do not state this axiomatic system here yet, but the reader can look in post 7 , the corresponding such axiomatic system not only for the real line, but for the material atomic euclidean geometry.

We must also notice here that the above axioms of the digital real line, do not support the division of quantities as this is possible by simple geometric constructions as the full system of 3-dimensional geometric axioms support (see e.g. post 7).

We describe here two different alternatives in constructing axiomatically the digital real numbers , the DIGITAL DECIMAL REAL NUMBERS (bottom-up construction, from smaller to larger), and the DIGITAL FRACTIONAL REAL NUMBERS (top-down construction, from larger to smaller) . These two classes of numbers systems are in general non-isomorphic, but there is appropriate sub-structure isomorphism , relative to a substructure of their exterior real numbers parts.
The digital real numbers are defined through the axioms of a linearly ordered field, with a small modification, and by adding some axioms about the resolution ω(r) or maximum high precision.

It is also discussed a finite inductive system

R(ω1)<R(ω2)....<R(ω3)<R(ωn)

of such digital real numbers as one only system of numbers with an levelness of precision.

We present here an axiomatic system of digital real numbers , with two only levels of precision , as this is the minimum number to formulate the Newton-Leibnitz calculus of functions.
But we could easily formulate an axiomatic system with n-levels of precision.
It is only in the context of the differential manifolds, and differential linear connections that we may need a 3-level precision system of real numbers.

We may notice in the next that contrary to the linearly ordered fields axioms of the real numbers system in the traditional mathematics with the infinite, here the real numbers without the infinite, can be better and more correctly introduced not through the operations but through axioms of the order <. the belonging ε, and of equality or congruence (under low or high precision) exactly as with the euclidean digital geometry and its lines (Eudoxus approach of ratios).

I do not claim here that the axioms of the digital natural numbers, below, are independent, in other words none of them can be proved from the others. As the elements are finite, there may be such a case. But I am strongly interested a) at first that are non-contradictory, and b) second that are adequate many, so as to describe the intended structure. later simplified and improved in elegance versions of the axioms may be given.

AXIOMS OF A SYSTEM OF TWO LEVELS OF PRECISION OF DIGITAL FRACTIONAL REAL NUMBERS AND DIGITAL DECIMAL REAL NUMBERS.

We follow here and modify the classical axioms of the real numbers from the book by M. Spivak , Calculus, chapter 28,"Construction of the real numbers", editions W.A. Benjamin Inc. 1957, and the book by H. A. Thurston “The number system” Dover 1956.
See also http://en.wikipedia.org/wiki/Real_number and
http://en.wikipedia.org/wiki/Construction_of_the_real_numbers

another interesting article is here

http://blogs.ams.org/matheducation/2015/03/10/one-reason-fractions-and-many-other-topics-are-hard-equivalence-relations-up-and-down-the-mathematics-curriculum/

The digital real numbers as system is denoted by R(ω(n),ω(r), Ω(r)) or simply R.
When we want to mean the set of classes of high precsion digital real number, in other words the Low precision digital real numbers, we use a bold R, to make the distinction.

The way to model them as a finite set, and thus have also the consistency of the axioms, and be utilizable for the digital euclidean geometry is the next.

Their ontology is (which can be modeled within the traditional real numbers )
1) A finite grid of its integers from -ω(n) to ω(n), with unit the unit 1 of the digital real numbers
2) A finite grid of its pixel-integers or Low precision visible points or external real numbers , from -ω(r) to ω(r)
with unit the pixel unit u1 (isomorphic to a segment of the digital integers again, but of smaller unit on the real axis)
3) A finite grid of its high pixel-integers or high precision invisible points or internal real numbers , from -Ω(r) to Ω(r), with high precision pixel unit u2
(isomorphic to a segment of the digital integers again, but of smaller unit on the real axis)
3.1) As alternative to the above, we may postulate a non-regular and non-homogeneous or non-equal-spaced grid from all + or - k/l * u1 , with 0< k*u1, l*u1<=ω(r). In that case we call these real numbers, digital fractional real numbers, as a contrast to the other alternative which is called digital decimal real numbers.

1) We have here an equivalence relation called Low precision equality (lp-equal) and symbolized by = , while the absolute equality is symbolized with 3-lines equality like identically equal. An equivalence relation has the properties of being reflexive a=a, Commutative if a=b then b=a, and Transitive if a=b and b=c, then a=c. It partitions the digital real numbers in to equivalence classes. The equivalence classes are called Low precision digital real numbers, while the elements of the equivalence classes High precision real numbers.

If ω(r) is the number of the Low precision digital real, we denote with Ω(r) the finite cardinal and ordinal number of the high precision real numbers.

And we have also two operation the addition + and the multiplication *.

We have also a subset of digital high precision real numbers denoted by P, and called positive real numbers.

We give then the finite decimal representation of the digital real numbers. ( see e.g. http://en.wikipedia.org/wiki/Decimal_representation ) But this is as far as the axioms of the digital real numbers will go (order, levels of precision and finite decimal representation). The usual properties of the operations are not included , in the axioms. The operations are defined with one of the systems of rounding, e.g. see http://en.wikipedia.org/wiki/Rounding or the IEEE system of floating rounding standards http://en.wikipedia.org/wiki/IEEE_floating_point_standard
http://en.wikipedia.org/wiki/Floating_point

The classical laws of the operations of the real numbers DO NOT HOLD for the digital real numbers.
E.g. because the equivalence relations of precision levels which are used as substitute of the equality is invariant to sufficient not very large translations, the law of calculation in addition holds. But it does not hold any cancellation law in multiplication or division because such transformation (multiplying and dividing) do not leave invariant the above equivalence relations.

The classical real numbers axioms (which are not included in the axioms of the digital decimal real numbers , as they may not hold exactly depending on the rounding system) are here.

2) (a+b)+c=a+(b+c) for all a, b and c in R.
3) There is a digital number 0 in R such that
3.1) a+0=a, for all a in R.
3.2) For every a in R there is some b in R such that
a+b=0. Such a, b is symbolized also by -a , and it is unique.
4) a+b=b+a for all a, b in R.
5) (a*b)*c=a*(b*c) for all a, b and c in R.
6) There is a digital number 1 in R not equal to 0, such that
6.1) a*1=a, for all a in R.
6.2) For every a in R not equal to 0, there is some b in R such that
a*b=1. Such a b is symbolized also by 1/a, and it is unique.
7) a*b=b*a for all a, b in R.
8) a*(b+c)=a*b+a*c for all a,b,c in R.

We have also a subset of digital high precision real numbers denoted by P, and called positive real numbers.
9) For all a in R, one and only one of the following 3 is true

9.1) a=0
9.2) a is in P
9.3) -a is in P (-a is the element such that a+(-a)=0 )

10) If a, b are in P, then a+b is in P
11) If a, b are in P, then a*b is in P

We define a high precision inequality a>b if a-b is in P
aa
a<=b if a= b if a> b or a=b

13) Here is the classical complete ordered field axiom.

Every non-empty subset of visible (or invisible) points which is bounded above has a least upper bound.

AN ALTERNATIVE WAY TO AXIOMATIZE THE DIGITAL DECIMAL REAL NUMBERS, IS TO START FROM FINITE DECIMAL REPRESENTATIONS OF THE REAL NUMBERS, UP TO SOME NUMBER OF DIGITS, IN DECIMALS AND UP TO SOME NUMBER OF DIGITS IN INTEGER PART, AS DEFINING THE 1ST PRECISION LEVEL OR RESOLUTION, AND THEN ALSO A 2ND GREATER NUMBER OF DECIMALS AS DEFINING THE 2ND PRECISION LEVEL.
TO DO SO WE NEED OF COURSE AN AXIOMATIC SYSTEM OF FINITE RATIONAL NUMBERS, SO AS TO DEFINE THE STANDARD FIELD OPERATIONS OF THE RATIONAL NUMBERS RE-PRESENTABLE IN DECIMAL NOTATION UP TO A 1ST AND 2ND DECIMAL PRECISION LEVEL. SUCH SYSTEM ALTHOUGH OF FINITE MANY ELEMENTS ARE BY NO MEANS CLOSED TO THE THE FIELD OPERATIONS.

THERE ARE 3 METHODSTO INTRODUCE THE DIGITAL REAL NUMBERS

1) THE CONSTRUCTIVE METHOD AFTER THE AXIOMATIC INTRODUCTION OF THE NATURAL NUMBERS.

(We define the digital integers, negative and positive then the digital rational numbers, and then the finite digital decimal represenations of the digital real numbers)

2) THE AXIOMATIC METHOD

(In this method we introduce the digotal real numbers from their finite decimal represenation at 4 precision levels and their order and operations)

3) THE DEFINITION METHOD AFTER THE AXIOMATIC SYSTEM OF THE CLASICAL REAL NUMBERS.

(This is the easiest method given the extended literature on the axiomatic system of the classical (analogue) real numbers with infinite many digits. We define after them the digital real numbers in such a way so that they exist in 4 precision levels and of only finite many decimal digits and we utilize only finite sets of them. )

In the next we follow the 3rd simplest method.

THE DEFINITION OF THE MULTI-PRECISION DECIMAL DIGITAL REAL NUMBERS R(n,m,q).

a) The rational numbers Q, as we known them, do involve the infinite, as they are infinite many, and are created with the goal in mind that proportions k/l of natural numbers k,l exist as numbers and are unique. The cost of course is that when we represent them with decimal representation they may have infinite many but with finite period of repetition decimal digits.

b) The classical real numbers R, as we know them, do involve the infinite, as they are infinite many, and are created with the goal in mind that proportions of linear segments of Euclidean geometry, exist as numbers and are unique (Eudoxus theory of proportions). The cost of course finally is that when we represent them with decimal representation they may have infinite many arbitrary different decimal digits without any repetition.

c) But in the physical or digital mathematical world, such costs are not acceptable. The infinite is not accepted in the ontology of mathematics (only in the subjective experience of the consciousness of the scientist). Therefore in the multi-precision digital real numbers, proportions are handled in different way, with priority in the Pythagorean idea of the creation of all numbers from an integral number of elementary units, almost exactly as in the physical world matter is made from atoms (here the precision level of numbers in decimal representation) and the definitions are different and more economic in the ontological complexity.

We will choose for all practical applications of the digital real numbers to the digital Euclidean geometry and digital differential and integral calculus, the concept of a system of digital decimal real numbers with three precision levels, lower, low and a high.

Definition 2.2 The definition of a PRECISION LEVEL P(n,m) where n, m are natural numbers , is that it is the set of all real numbers that in the decimal representation have not more than n decimal digits for the integer part and not more than m digits for the decimal part. Usually we take m=n. In other words as sets of real numbers it is a nested system of lattices each one based on units of power of 10, and as union a lattice of rational numbers with finite many decimal digits. We could utilize other bases than 10 e.g. 2 or 3 etc, but for the sake of familiarity with the base 10 and the 10 fingers of our hands we leave it as it is.

THE DEFINITION (AS A KIND OF AXIOMS) OF THE DIGITAL REAL NUMBERS R(n,m,q)

We assume at least three precision levels for an axiomatic decimal system of digital real numbers R(n,m,q) : THE LOCAL LOWER PRECISION LEVEL P(n) , THE LOW PRECISION LEVEL P(m) , AND THE HIGH PRECISION LEVEL P(q). Each precision level of order k has 10^(4k+2log2) points or numbers where by log we denote the logarithm with base 10. It has 10^k positive decimal numbers , which are doubled for the negative ones, thus in total at most 2*(10^(2k)) <= 10^(2k+log2). And again so many for the integer numbers, thus in total 10^(4k+2log2)). Now if for 3-dimensional geometric applications as coordinates of points, this will give 10^(12k+6log2)) points inside a big cube.

Whenever we refer to a real number x of a (minimal in precision levels) system of real numbers R(n,m,q) , we will always mean that x belongs to the local lower precision level P(n) and that the system R(n,m,q) has at least three precision levels with the current axioms.

Whenever we write an equality relation =_m we must specify in what precision level it is considered. The default precision level that a equality of numbers is considered to hold, is the low or standard precision level P(n).

Some of the Linearly ordered Field operations

The field operations in a precision level are defined in the usual way, from the decimal representation of the numbers. This would be an independent definition, not involving the infinite. Also equality of two numbers with finite decimal digits should be always specified to what precision level. E.g. if we are talking abut equality in P(m) we should symbolize it my =_m, while if talking about equality in P(q) we should symbolize it by =_q .If we want to define these operation from those of the real numbers with infinite many decimal digits, then we will need the truncation function [a]x of a real number a , in the Precision level P(x).

Then the operations e.g. in P(n) with values in P(m) n<<m would be

[a]_n+[b]_n=_m[a+b]_m(eq. 3)

[a]_n* [b]_n=_m[a*b]_m(eq. 4)

([a]_n)^(-1)=_m[a^(-1)]_m(eq. 5)

(Although, the latter definition of inverse seems to give a unique number in P(m), there may not be any number in P(m) or not only one number in P(m), so that if multiplied with [a]_nit will give 1. E.g. for n=2 , and m=5 , the inverse of 3, as ([3]_n )^(-1)=_m[1/3]_m =0.33333 is such that still 0.33333*3≠_m1 ).

Such a system of double or triple precision digital real numbers, has closure of the linearly ordered field operations only in a specific local way. That is If a, b belong to the Local Lower precision, then a+b, a*b , -a, a^(-1) belong to the Low precision level, and the properties of the linearly ordered commutative field hold: (here the equality is always in P(m), this it is mean the =_m).

1) if a, b, c belong to P(n) then (a+b), (b+c), (a+b)+c, a+(b+c) belong in P(m) and

(a+b)+c=a+(b+c) for all a, b and c in P(m).

2) There is a digital number 0 in P(m) such that

2.1) a+0=a, for all a in P(m).

2.2) For every a in P(n) there is some b in P(m) such that

a+b=0. Such a, b is symbolized also by -a , and it is unique in P(m).

3) if a, b, belong to P(n) then (a+b), (b+a), belong in P(m) and

a+b=b+a

4) if a, b, c belong to P(n) then (a*b), (b*c), (a*b)*c, a*(b*c) belong in P(m) and

(a*b)*c=a*(b*c).

5) There is a digital number 1 in P(m) not equal to 0 in P(m), such that

5.1) a*1=a, for all a in P(m).

5.2) For every a in P(n) not equal to 0, there may be one or none or not only one b in P(m) such that a*b=1 . Such b is symbolized also by 1/a, and it may not exist or it may not be unique in P(m).

6) if a, b, belong to P(n) then (a*b), (b*a), belong in P(m) and

a*b=b*a

7) if a, b, c belong to P(n) then (b+c) , (a*b), (a*c), a*(b+c), a*b+a*c, belong in P(m) and

a*(b+c)=a*b+a*c

Which numbers are positive and which negative and the linear order of digital numbers is precision levels P(n), P(m), P(q) is something known from the definition of precision levels in the theory of classical real numbers in digital representation.

If we denote by PP(n) the positive numbers of P(n) and PP(m) the positive numbers of P(m) then

8) For all a in PP(n), one and only one of the following 3 is true

8.1) a=0

8.2) a is in PP(n)

8.3) -a is in PP(n) (-a is the element such that a+(-a)=0 )

9) If a, b are in PP(n), then a+b is in PP(m)

10) If a, b are in PP(n), then a*b is in PP(m)

It holds for the inequality a>b if and only if a-b is in PP(m)

aa

a<=b if a<b or a=b

a>= b if a> b or a=b

and similar for PP(m).

Similar properties as the ones from P(n) to P(m) hold if we substitute n with m, and m with q.

Also, the Archimedean property holds only recursively in respect e.g. to the local lower precision level P(n).

In other words, if a, b, ab. And similarly for the precision levels P(m) and P(q).

The corresponding to the Eudoxus-Dedekind completeness in the digital real numbers also is relative to the three precision levels.

We define that two visible points A, B, are in contact or of zero distance distance(A,B)=0, if and only if there are invisible points A’ belonging to A (see axioms of incidence) and B’ belonging to B, so that distance(A’.B’)<=1/(10^q). Two visible points in contact do not have in general the same Cartesian measures distance (see Definition 2.3.I.2 )

In other words for every visible point A in the Low precision level , there are exactly two other points B1, B2 again in the Low precision level with B1<A<B2 , such that the distance between A and B1, and A, B2 is zero in the Low precision level, and there is no other visible point C strictly between A and B1 and a and B2. This can be derived also from the requirement that all possible combinations of decimal digits in the local lower, low and high precision levels are being used as numbers of the system of digital real numbers.

Sufficient Mutual inequalities of the precision levels

We impose also axioms for the sufficiently large size of the high precision level relative to the other two, and the sufficient large size of the low precision level relative to the local lower precision level. That is for the mutual relations of the integers m, n, q.

It may seem that these differences of the resolution or the precision levels are very severe and of large in between distance, and not really necessary. It may be so, as the future may show. But for the time being we fell safe to postulate such big differences.

1) If we repeat the operations of addition and multiplication of the linearly ordered commutative field starting from numbers of the local lower precision level P(n), so many times as the numbers of the local lower precision level P(n), then the results are still inside the low precision level P(m). (This in particular gives that (10^n)^(10^n)<=(10^m)). This is also expressed by saying that the 10^m is seemingly infinite compared to the 10^n.

2) The largest error in the high precision level P(q), which we may also identify as the smallest magnitude in the low precision level P(m) in other words the 10^(-m), will appear as zero error in the low precision level P(n), even after additive repetitions that are as large as the cardinal number of points of the lower precision level P(n). This is e.g. is guaranteed if 5n+2log2<m or rounded 6n<m (Where by log we denote the logarithm with base 10) . The points in 1-dimensional geometry are 10^(4n+2log2) and if an error of order 10^(-m) is repeated so many times and still be less than 10^(-n), then 10^(4n+2log2)*10^(-m)<10^(-n), thus 5n+2log2<m. For the Euclidean geometry cube, this requires that 10^(12n+6log2)*10^(-m)<10^(-n) thus 13n+6log2<m or rounded 14n<m. This may be expressed also by saying that the 10^(-m) is seemingly infinitesimal compared to the 10^(-n).

3) The smallest magnitude in the high precision level P(q) in other words the 10^(-q), will appear as zero error in the low precision level P(m), even after additive repetitions as large as the cardinal number of points of the low precision level P(m). This is e.g. guaranteed if 5m+2log2<q or rounded 6m<q, and for Euclidean geometry applications 13m+6log2<q or rounded 14m<q.This may be expressed also by saying that the 10^(-q) is seemingly infinitesimal compared to the 10^(-m).

If instead of three precision levels P(n), P(m), P(q), we would introduce four precision levels (still another (P(r)), with the same mechanism of recursive axioms, then we would denote it by R(n,m,q,r) and we would call it a 4-precisions levels system of digital real numbers.

Two digital systems of Real numbers R(n,m,q) , R(n’,m’,q’) with n=n’, m=m’, q=q’ and the above axioms are considered isomorphic.

SEEMINGLY IRRATIONAL NUMBERS WITHIN THE DIGITAL REAL NUMBERS

The seemingly irrational numbers can be defined as the digital real numbers that have a seemingly infinite number of digits in their decimal representation and cannot be written as quotients of two computable integers (or non-seemingly infinite integers). It is interesting to investigate the the digital real numbers that have a seemingly infinite number of digits in their decimal representation and if there is a period in their decimal representation this period is of a non-computable integer (or non-seemingly infinite integer).

We should point out that in the next considerations, the digital rational numbers, are different from the classical rational numbers, as they do not include classical rational numbers with decimal represation which is of infinite periodic decimal digits. Therefore not onlt the classical irrational numbers but also some classical rational numbers, are in the digital real numbers semingly irrational numbers.

THE LONGER DEFINITION OF THE NATURAL OR DIGITAL REAL NUMBERS WITH 4 PRECISION LEVELS

THE MULTI-PRECISION DECIMAL DIGITAL REAL NUMBERS R(m,n,p,q)

Rules for phantasy and drawing of figures.

m<n<p<q.

c) But in the physical or digital mathematical world, such costs are not acceptable. The infinite is not accepted in the ontology of digital mathematics (only in the subjective experience of the consciousness of the scientist). Therefore in the multi-precision digital real numbers, proportions are handled in different way, with priority in the Pythagorean-Democritus idea of the creation of all numbers from an integral number of elementary units, almost exactly as in the physical world matter is made from atoms (here the precision level of numbers in decimal representation) and the definitions are different and more economic in the ontological complexity.

Definition 1.1 The definition of a PRECISION LEVEL P(n,m) where n, m are natural numbers , is that it is the set of all real numbers that in the decimal representation have not more than n decimal digits for the integer part and not more than m digits for the decimal part. Usually we take m=n. In other words as sets of real numbers it is a nested system of lattices each one based on units of power of 10, and as union a lattice of rational numbers with finite many decimal digits. We could utilize other bases than 10 e.g. 2 or 3 etc, but for the sake of familiarity with the base 10 and the 10 fingers of our hands we leave it as it is.

It is direct that an estimate of the cardinal number, symbolized by |P(m)|, of a precision level

P(m)=P(m,m), is at most twice (decimal and integer part) all the repetitive combinations of 10 by m, thus |P(m)|<=2*10^m.

THE 4 PRECISION LEVELS-RESOLUTIONS

P(m): for length measures, volumes, areas. Visible points

P(n): visible points of negligible, seemingly infinitesimal sizes, measures.

P(p): Invisible points-pixels of negligible seemingly infinitesimal sizes, measures.

P(q): Invisible points-atoms of negligible seemingly infinitesimal sizes, measures.

As initially we considered a system of digital real numbers R(m,n,p,q) we consider the points of P(m), P(n) as visible in the figures while the points of P(p) as invisible pixels , and those of P(q) as invisible atoms. Therefore, even the points and seemingly infinitesimals that will be defined below, of P(n) relative to P(m) are considered as visible. This is in accordance with the habit in classical mathematics to make the points visible, although they claim that they have zero size.

THE DEFINITION 1OF THE NATURAL OR DIGITAL REAL NUMBERS R(m,n,p,q) WITH 4 PRECISION LEVELS

We assume at least four precision levels for an axiomatic decimal system of digital real numbers
Whenever we refer to a real number x of a (minimal in precision levels) system of real numbers R(m,n,p,q) , we will always mean that x belongs to the precision level P(m) and that the system R(m,n,p,q) has at least four precision levels with the current axioms.

Some of the Linearly ordered Field operations

The field operations in a precision level are defined in the usual way, from the decimal representation of the numbers. This would be an independent definition, not involving the infinite. Also equality of two numbers with finite decimal digits should be always specified to what precision level. E.g. if we are talking about equality in P(m) we should symbolize it my =_m, while if talking about equality in P(q) we should symbolize it by =_q .If we want to define these operation from those of the real numbers with infinite many decimal digits, then we will need the truncation function [a]x of a real number a , in the Precision level P(x). Here for the rounding function we use the rounding to the left for positive numbers and to the right for negative numbers.

Then the operations e.g. in P(m) with values in P(n) m<<n would be

[a]_m+[b]_m=_n[a+b]_n(eq. 3)

[a]_m* [b]_m=_n[a*b]_n(eq. 4)

([a]_m)^(-1)=_n[a^(-1)]_n(eq. 5)

(Although, the latter definition of inverse seems to give a unique number in P(n), there may not be any number in P(n) or not only one number in P(n), so that if multiplied with [a]_mit will give 1. E.g. for m=2 , and n=5 , the inverse of 3, as ([3]_m )^(-1)=_n[1/3]_n =0.33333 is such that still 0.33333*3≠_n1 ).

Nevertheless here we will not involve the infinite and the classical real numbers, and we take the operation of digital real numbers from the standard operations of them as numbers with finite digital decimal representation and truncation by rounding.

Such a system of double or triple precision digital real numbers, has closure of the linearly ordered field operations only in a specific local way. That is If a, b belong to the Local Lower precision, then a+b, a*b , -a, a^(-1) belong to the Low precision level, and the properties of the linearly ordered commutative field hold: (here the equality is always in P(n), this it is mean the =_n).

1) if a, b, c belong to P(m) then (a+b), (b+c), (a+b)+c, a+(b+c) belong in P(n) and

(a+b)+c=a+(b+c) for all a, b and c in P(n).

2) There is a digital number 0 in P(n) such that

2.1) a+0=a, for all a in P(n).

2.2) For every a in P(m) there is some b in P(n) such that

a+b=0. Such a, b is symbolized also by -a , and it is unique in P(n).

3) if a, b, belong to P(m) then (a+b), (b+a), belong in P(n) and

a+b=b+a

4) if a, b, c belong to P(m) then (a*b), (b*c), (a*b)*c, a*(b*c) belong in P(n) and

(a*b)*c=a*(b*c).

5) There is a digital number 1 in P(n) not equal to 0 in P(n), such that

5.1) a*1=a, for all a in P(n).

5.2) For every a in P(m) not equal to 0, there may be one or none or not only one b in P(n) such that a*b=1 . Such b is symbolized also by 1/a, and it may not exist or it may not be unique in P(n).

6) if a, b, belong to P(m) then (a*b), (b*a), belong in P(n) and

a*b=b*a

7) if a, b, c belong to P(n) then (b+c) , (a*b), (a*c), a*(b+c), a*b+a*c, belong in P(n) and

a*(b+c)=a*b+a*c

Which numbers are positive and which negative and the linear order of digital numbers is precision levels P(m), P(n), etc is something known from the definition of precision levels in the theory of classical real numbers in digital representation.

If we denote by PP(m) the positive numbers of P(m) and PP(n) the positive numbers of P(n) then

8) For all a in PP(m), one and only one of the following 3 is true

8.1) a=0

8.2) a is in PP(m)

8.3) -a is in PP(m) (-a is the element such that a+(-a)=0 )
9) If a, b are in PP(m), then a+b is in PP(n)

10) If a, b are in PP(m), then a*b is in PP(n)
It holds for the inequality a>b if and only if a-b is in PP(n)

aa

a<=b if a<b or a=b

a>= b if a> b or a=b

and similar for PP(n).

Similar properties as the ones from P(m) to P(n) hold if we substitute n with m, and m with p, q.

For the R(m,n) the integers of P(m) are also called computable finite or countable finite, while those of P(n) are unaccountable finite or non-computable finite or also seemingly infinite relative to P(m).

Also, the Archimedean property holds only recursively in respect e.g. to the local lower precision level P(m).

In other words, if a, b, ab. And similarly for the precision levels P(n) and P(p),P(q).

The corresponding to the Eudoxus-Dedekind completeness in the digital real numbers also is relative to the three precision levels is simply that in the precision levels all possible combination of digits are included and not any decimal number of P(m) or P(n) is missing. Still this gives

THE SUPREMUM COMPLETENESS PROPERTY OF THE DIGITAL REAL NUMBERS.

From this completeness we deduce the supremum property of upper bounded sets (and infimum property of lower bounded sets) in the P(m) (but also P(n)) precision levels. This is because in well ordered sets holds the supremum property of upper bounded sets. Here lower bounded sets have also the infimum property and this holds for any resolution of the digital real numbers
Mutual inequalities of the precision levels (AXIOMS OF SEEMINGLY (m,n) -INFINITE OR (m,n)-UNCOUNTABLE OR NON-COMPUTABLE FINITE AMONG RESOLUTIONS and seemingly finite or visibly finite or bounded or computable finite numbers. )

There are definitions modeled after the definitions of inaccessible cardinals in classical mathematics. Here we give a weaker alternative definitions with weaker concepts of seemingly infinite that would correspond to that of inaccessible cardinals. In other words we do not include the operation of power.

We may conceive the countable finite as a finite computable by an appropriate computational power of some computer, and unaccountable finite as the finite not computable by an appropriate power of a computer

Transcendental Orders of (m,n) seemingly infinite, as in classical mathematics transcendental orders of ordinal numbers are also definable. E.g. if a, b are (m,n)=seemingly infinite then a is transcendental larger than b, in symbols a>>b iff b/a=_m0 in P(m).

And similarly transcendental orders of seemingly infinitesimals. E.g. if a, b are (m,n)=seemingly infinitesimals then a is transcendental smaller than b, in symbols a<<b iff a/b=_m0 in P(m).

We may compare them with the small o() and big O() definitions of the classical mathematics, but they are different as the latter involve the countable infinite, while former here involve only finite sets of numbers.

9) REQUIREMENTS OF THE SEEMINGLY INFINITE If we repeat the operations of addition and multiplication of the linearly ordered commutative field starting from numbers of the precision level P(m), so many times as the numbers of the local lower precision level P(m), then the results are still inside the low precision level P(n). In symbols if by |P(m)| we denote the cardinality of P(m), (we know from the beginning of the paragraph that thus |P(m)|<=2*10^m.) then the definition would hold if,

|P(m)|*(10^m), and (10^m)^|P(m)|<=10^n . Similarly for the pair (m,q), because the largest element of P(m) is of course the 10^m,and the largest element of P(n) is the 10^n. We may express it by saying that the 10^n is seemingly infinite or unaccountable finite compared to 10^m, or that the numbers less than 10^m are countable or computable finite. If we include besides the addition and multiplication the power operation too, then 10^n is inaccessible seemingly infinite compared to 10^m (a concept similar to inaccessible cardinal numbers in classical mathematics). Similarly, we may define the relevant terms for the precision levels P(p), P(q).

10) REQUIREMENTS OF THE SEEMINGLY INFINITESIMALS The smallest magnitude in the precision level P(n) in other words the 10^(-n), will appear as zero error in the precision level P(m), even after additive repetitions that are as large as the cardinal number of points of the lower precision level P(m) and multiplied also by any large number of P(m). In symbols

10^(-n)*|P(m)|*10^m<=10^(-m). Similarly for the pairs (n,p), (p,q).

We may re-write the inequality as 10^(m)*|P(m)|*10^m<=10^(n). This is certainly im;ied by the inequality stated in the 9) about the seemingly infinite (10^m)^|P(m)|<=10^n

Thus the requirements of seemingly infinitesimals of the pair P(m)/P(n) follows from the requirement of P(n) being seemingly infinite compared to the P(m).

This may also be expressed by saying that the 10^(-n) is seemingly infinitesimal compared to the 10^(-m) . Other elements of P(n) symbolized by dx with |dx|<10^-m with the same inequalities, that is |dx|*|P(m)|*10^m<=10^(-m) are also seemingly infinitesimals.

We may repeat such definitions and requirements for the pairs P(n)/P(p), P(p)/P(q).

The next requirements are required also to be met for the relation of seemingly infinitesimals P(p) relative to P(m):

The seemingly infinitesimals dx of P(p) relative to P(m) (thus |dx|<=10^(-m) ) are required in the proofs that will follow in this work to have properties that resemble the ideals in ring theory (see e.g. VAN DER WAERDEN ALGEBRA Vol I, chapter 3, Springer 1970 ). More precisely what it is required to hold is that:

If a, b are elements of P(m) , and dx dy seemingly infinitesimals of P(p) relative to P(n) (thus |dx|, |dy| <=10^(-n) ,thus relative to P(m) too) then the linear combination and product are still seemingly infinitesimals if repeated as many times as the cardinality of P(m). In symbols adx+bdy , are seemingly infinitesimals of P(n) relative to P(m) and dx*dy is seemingly infinitesimal of P(q) relative to P(p) and thus relative to P(m) too.

This requirement though is a direct consequence of the definition of seemingly infinitesimals of the pairs P(m)/P(n). P(n)/P(p), P(p)/P(q).

We call this the ideal-like property of the seemingly infinitesimals.

One very important equation is of course that the digital real numbers is the union of the four precision levels.

R(m, n, p, q)= P(m)ᴗP(n)ᴗP(p) ᴗP(q)

Two digital systems of Real numbers R(m,n,p,q) , R(m’,n’,p’,q’) with m=m’, n=n’, p=p’,q=q’ and the above axioms are considered isomorphic.

SEEMINGLY IRRATIONAL NUMBERS WITHIN THE DIGITAL REAL NUMBERS

The seemingly irrational numbers can be defined as the digital real numbers that have a seemingly infinite number of digits in their decimal representation and cannot be written as quotients of two computable integers (or non-seemingly infinite integers). It is interesting to investigate the the digital real numbers that have a seemingly infinite number of digits in their decimal representation and if there is a period in their decimal representation this period is of a non-computable integer (or non-seemingly infinite integer).

We should point out that in the next considerations, the digital rational numbers, are different from the classical rational numbers, as they do not include classical rational numbers with decimal represation which is of infinite periodic decimal digits. Therefore not onlt the classical irrational numbers but also some classical rational numbers, are in the digital real numbers semingly irrational numbers.

AXIOMS OF A SYSTEM OF THREE LEVELS OF PRECISION OF DIGITAL FRACTIONAL REAL NUMBERS AND DIGITAL DECIMAL REAL NUMBERS.

We follow here and modify the classical axioms of the real numbers from the book by M. Spivak , Calculus, chapter 28,"Construction of the real numbers", editions W.A. Benjamin Inc. 1957, and the book by H. A. Thurston “The number system” Dover 1956.
See also http://en.wikipedia.org/wiki/Real_number and
http://en.wikipedia.org/wiki/Construction_of_the_real_numbers

The digital real numbers as system is denoted by R(ω(n),ω(r), Ω1(r) ,Ω2(r)) or simply R.
When we want to mean the set of classes of highest precision digital real number, in other words the Low precision digital real numbers, we use a bold R, to make the distinction.
1) We have here two equivalence relation called Low precision equality (lp-equal) and symbolized by = , and High precision equality (hp-equal) symbolized by =(h) while the absolute equality is symbolized with 3-lines equality like identically equal. An equivalence relation has the properties of being reflexive a=a, Commutative if a=b then b=a, and Transitive if a=b and b=c, then a=c. It partitions the digital real numbers in to equivalence classes. The equivalence classes of the low precision over the high precsion equivalence classes are called Low precision digital real numbers, while the equivalence classes of the High precision equivalence relation over the highest precision numbers is called High precision real numbers.
The mappings from the high precision digital real numbers to the low precision digital numbers that they belong, are homomorphisms relative to the order (less or equal) ,
If ω(r) is the number of the Low precision digital real, we denote with Ω1(r) the finite cardinal and ordinal number of the high precision real numbers, and by Ω2(r) the finite cardinal and ordinal number of the highest precision real numbers,

And we have also two operation the addition + and the multiplication *. Such operation are operations on the highest precision digital real numbers. (the operations are defined on the highest precision only in the axioms)

Their axiomatic ontology is
(which it can be modeled within the traditional real numbers )
1) A finite grid of its integers from -ω(n) to ω(n), with unit the unit 1 of the digital real numbers.
2) A finite grid of its pixel-integers or Low precision visible points or external real numbers, from -ω(r) to ω(r), with low pixel unit u1
(isomorphic to a segment of the integers again, but of smaller unit on the real axis)
3) A finer finite grid of its pixel-integers or High precision invisible points or intermediate real numbers , from -Ω1(r) to Ω1(r), with finer pixel unit u2
(isomorphic to a segment of the digital integers again, but of smaller unit on the real axis)
3.1) As alternative to the above, we may postulate a non-regular and non-homogeneous or non-equal-spaced grid from all + or - (k/l)* u1, with 0< k*u1, l*u1<=ω(r). In that case we call these real numbers, digital fractional real numbers, as a contrast to the other alternative which is called digital decimal real numbers.

4) A finer finite grid of its pixel-integers or Highest precision invisible points or internal real numbers , from -Ω2(r) to Ω2(r), with unit u2
(isomorphic to a segment of the digital integers again, but of smaller unit on the real axis)
4.1) As alternative to the above, we may postulate a non-regular and homogeneous grid from all + or - (k/l) , with 0< k, l in the grid of high precision numbers. In that case we call these real numbers, digital fractional real numbers, as a contrast to the other alternative which is called digital decimal real numbers.

We have also a subset of digital highest precision real numbers denoted by P, and called positive real numbers.

We give then the finite decimal representation of the digital real numbers.( see e.g. http://en.wikipedia.org/wiki/Decimal_representation ) But this is as far as the axioms of the digital real numbers can go (order, levels of precision and finite decimal representation) . The usual properties of the operations are not included , in the axioms. The operations are defined with one of the systems of rounding, e.g. see http://en.wikipedia.org/wiki/Rounding or the IEEE system of floating rounding standards http://en.wikipedia.org/wiki/IEEE_floating_point_standard and http://en.wikipedia.org/wiki/Floating_point

The classical laws of the operations of the real numbers DO NOT HOLD for the digital real numbers.
E.g. because the equivalence relations of precision levels which are used as substitute of the equality is invariant to sufficient not very large translations, the law of calculation in addition holds. But it does not hold any cancellation law in multiplication or division because such transformation (multiplying and dividing) do not leave invariant the above equivalence relations.

The classical axioms of the real numbers (which are not included in the axioms of the digital decimal real numbers , as they may not hold exactly depending on the rounding system ) are

2) (a+b)+c=a+(b+c) for all a, b and c in R.
3) There is a digital number 0 in R(ω) such that
3.1) a+0=a, for all a in R.
3.2) For every a in R(ω) there is some b in R such that
a+b=0. Such a, b is symbolized also by -a , but it is only lp-equality unique.
4) a+b=b+a for all a, b in R.
5) (a*b)*c=a*(b*c) for all a, b and c in R.
6) There is a digital number 1 in R not lp-equal to 0, such that
6.1) a*1=a, for all a in R.
6.2) For every a in R not lp-equal to 0, there is some b in R such that
a*b=1. Such a b is symbolized also by 1/a, but it is only lp-equality unique.
7) a*b=b*a for all a, b in R.
8) a*(b+c)=a*b+a*c for all a,b,c in R.

9) For all a (highest precision digital numbers ) in R, one and only one of the following 3 is true

9.1) a=0
9.2) a is in P
9.3) -a is in P (-a is the element such that a+(-a)=0 )

10) If a, b are in P, then a+b is in P
11) If a, b are in P, then a*b is in P

We define a highest precision inequality a>b if a-b is in P
aa
a<=b if a= b if a> b or a=b

And we also define a High and Low precision inequality by a<b . Let us give the definition of the low precision order. It holds that a<b if and only if all high precision digital real numbers belonging to a, are smaller than all high precision digital real numbers belonging to b. And similarly for the high precision inequality defined from the highest precision inequality.

12) In the digital real numbers we want that there is an absorbing elements ω , such that for all a in P
12.1) a+ω=ω

And it holds that for all a in R , -ω<=a<= ω

13) Here is the classical complete ordered field axiom.

Every non-empty subset of highest precision (or high or low precision) numbers which is bounded above has a least upper bound.

14) There is one more axiom relating the absolute equality of the digital real numbers and the lp-equality. It is stated as a relation between the cardinal number of low precision or visible real numbers and high precision or invisible real numbers. Let us denote also by ω(n) the size of the natural numbers defined by the unit 1 of the real numbers inside the system.
By ω(r) is denoted the finite cardinal of the low precision digital real numbers, and by Ω1(r) the finite cardinal number of the high precision digital real numbers and by Ω2(r) the finite cardinal number of the highest precision digital real numbers.

We call it the continuum axiom or sufficient high resolution axiom or sufficient high density axiom:

ω(n)<= ω(r)
ω(r)<= Ω1(r)
Ω1(r)<= Ω2(r)

For the digital fractional real numbers, we have the option to postulate the Ω1(r) , sufficient large compared to the ω(r), so that all classical operations of addition, subtraction, multiplication and division of two low precision numbers (or external real numbers) have exact outcome with values in the high precision numbers (or intermediate real numbers). And similarly the Ω2(r) so large compared to the Ω1(r), so that all classical operations of addition, subtraction, multiplication and division of two high precision numbers (or intermediate real numbers) have exact outcome with values in the highest precision numbers (or internal real numbers).

The new axiomatic system (as the old infinite real numbers too) is a categorical axiomatic system for each ω(n), ω(r) and Ω1(r) Ω2(r). In other words any two models Μ1, Μ2 of the digital real numbers of 3-levels of precision of the same sizes ω(n), ω(r) and Ω1(r) Ω2(r) are isomorphic. The tetrad ω(n), ω(r) and Ω1(r) Ω2(r) defines up to isomorphism the system of the digital real numbers R, that is why it is also denoted by R(ω(n),ω(r), Ω1(r),Ω2(r)) . (It is be noticed that there are no infinite sets, in the digital set theory, thus all models of the above axiomatic system will be finite sets)

THE DIGITAL ORDINAL ( OR TREE ) REAL NUMBERS

We describe in the next an ingenious simple top-down method to construct the digital fractional real numbers , through only their order and , middle numbers.

(to be continued)

ABOUT THE NUMBERS e AND π or rather....THE ALGORITHM e AND THE ALGORITHM π.

In the digital real numbers the "numbers" e and π are not really numbers, but only algorithms of finite syntax. They become numbers only if we stop at a finite number of digits. So strictly speaking we should talk about the algorithm e, and the algorithm π

References

H. A. Thurston “The number system” Dover 1956

Pure mathematics G.H hardy

J.H. Conway “On Numbers and Games” Academic Press 1976

1) Rozsa Peter “Playing with Infinity” Dover Publications 1961

2) R. L. Wilder “Evolution of mathematical Concepts” Transworld Publishers LTD 1968

3) Howard Eves “An Introduction to the History of Mathematics”,4^th edition 1953 Holt Rinehart and Winston publications

4) Howard Eves “Great Moments in Mathematics” The Mathematical Association of America 1980

5) Hans Rademacher-Otto Toeplitz “The Enjoyment of Mathematics”Princeton University Press 1957.

6) R. Courant and Herbert Robbins “What is Mathematics” Oxford 1969

7) A.D. Aleksndrov, A.N. Kolmogorov, M.A. Lavrentev editos

“Mathematics, its content, methods, and meaning” Vol 1,2,3 MIT press 1963

8) Felix Kaufmann “The Infinite in Mathematics” D. Reidel Publishing Company 1978

9) Edna E. Kramer “The Nature and Growth of Modern Mathematics”

Princeton University Press 1981

10) G. Polya “Mathematics and plausible reasoning” Vol 1, 2 1954 Princeton University press

11) Maurice Kraitchik “Mathematique des Jeux” 1953 Gauthier-Villars

12) Heinrich Dorrie “100 Great Problems of Elementary Mathematics”

Dover 1965

13) Imre Lakatos “Proofs and Refutations” Cambridge University Press 1976

“Dtv-Atlas zur Mtahematik” Band 1,2,1974

14) Struik D. J. A Concise History of Mathematics Dover 1987

15) S. Bochner The Role of Mathematics in the Rise of Science Princeton 1981

16) D.E. Littlewood “Le Passé-Partout Mathematique” Masson et c, Editeurs Paris 1964

17) A concise history of mathematics , by D. J. Struik, Dover 1987

18) Cohen, forcing method

Book about the continuum hypothesis.

At the end of this chapter, there is

a) Advantages-disadvantages of these new digital mathematics compared to the classical analogue, infinitary mathematics.

b) A fictional discussion in dialogue form of celebrated historic creators of the old mathematics praising or questioning the new mathematics compared to the old.

Speakers

Pythagoras, Euclid, Eudoxus, Newton, Leibniz, Dedekind, Cauchy, Weierstrass, Cantor, Kyr, von Neuman etc

The new Fair Natural Mathematics without the infinite, of the 2nd Millennium.

Thursday, January 30, 2014

6. The Axiomatic multi-precision decimal Digital or Democritus Real Numbers.