General Information Framework

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General Information Framework (GIF) set theory


Set (which notated by "{" and "}") is an object that used as General Information Framework.

Set's property depends on its information's type.

There are 2 basic types of information that can be examined through GIF.

1.a) Empty set ={}
2.a) Non-empty set

(2.a) has 3 non-empty set's types:

1.b) Finitely many objects ( {a,b} ).
2.b) Infinitely many objects ( {a,b,...} ).
3.b) Infinite object ( {__} ).

(3.b) Is the opposite of the Empty set, therefore {__}=Full set.

GIF has two limits:

The lowest limit is {}(=Empty set).

The highest limit is {__}(=Full set).

Both limits are unreachable by (1.b) or (2.b) non-empty set's types.

Or in another words:

Any information system exists in the open interval of ({},{__}).


Infinitely many objects ( {a,b,...} ) cannot be completed, therefore words like 'all' or 'complete' cannot be used with sets that have infinitely many objects.

{} or {__} are actual infinity.

{a,b,...} is potential infinity.

An example:

http://www.geocities.com/complementarytheory/LIM.pdf


Question:

So what can we do with this theory that we can't do with standard set theory?

A non-formal answer (yet):

Please look at this two articles:

http://www.geocities.com/complementarytheory/ET.pdf

http://www.geocities.com/complementarytheory/CATheory.pdf


At this stage you have to look at them as non-formal overviews, but with a little help from my friends, they are going to be addressed in a rigorous formal way.

All the energy that was used to research the transfinite universes, is going to be used to research the information concept itself, including researches that explore our own cognition's abilities to create and develop the Math language itself.

By GIF set theory our models does not have to be quantified before we can deal with them, because GIF set theory has the ability to deal with any information structure in a direct way, which keeps its dynamic natural complexity during the research.

Concepts like symmetry-degree, Information's clarity-degree, uncertainty, redundancy and complementarity, are some of the fundamentals of this theory.


Organic
 
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The first problem of Hilbert

1. Cantor's problem of the cardinal number of the continuum
Two systems, i. e, two assemblages of ordinary real numbers or points, are said to be (according to Cantor) equivalent or of equal cardinal number, if they can be brought into a relation to one another such that to every number of the one assemblage corresponds one and only one definite number of the other. The investigations of Cantor on such assemblages of points suggest a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving. This is the theorem:

Every system of infinitely many real numbers, i. e., every assemblage of numbers (or points), is either equivalent to the assemblage of natural integers, 1, 2, 3,... or to the assemblage of all real numbers and therefore to the continuum, that is, to the points of a line; as regards equivalence there are, therefore, only two assemblages of numbers, the countable assemblage and the continuum.

From this theorem it would follow at once that the continuum has the next cardinal number beyond that of the countable assemblage; the proof of this theorem would, therefore, form a new bridge between the countable assemblage and the continuum.

Let me mention another very remarkable statement of Cantor's which stands in the closest connection with the theorem mentioned and which, perhaps, offers the key to its proof. Any system of real numbers is said to be ordered, if for every two numbers of the system it is determined which one is the earlier and which the later, and if at the same time this determination is of such a kind that, if a is before b and b is before c, then a always comes before c. The natural arrangement of numbers of a system is defined to be that in which the smaller precedes the larger. But there are, as is easily seen infinitely many other ways in which the numbers of a system may be arranged.

If we think of a definite arrangement of numbers and select from them a particular system of these numbers, a so-called partial system or assemblage, this partial system will also prove to be ordered. Now Cantor considers a particular kind of ordered assemblage which he designates as a well ordered assemblage and which is characterized in this way, that not only in the assemblage itself but also in every partial assemblage there exists a first number. The system of integers 1, 2, 3, ... in their natural order is evidently a well ordered assemblage. On the other hand the system of all real numbers, i. e., the continuum in its natural order, is evidently not well ordered. For, if we think of the points of a segment of a straight line, with its initial point excluded, as our partial assemblage, it will have no first element.

The question now arises whether the totality of all numbers may not be arranged in another manner so that every partial assemblage may have a first element, i. e., whether the continuum cannot be considered as a well ordered assemblage—a question which Cantor thinks must be answered in the affirmative. It appears to me most desirable to obtain a direct proof of this remarkable statement of Cantor's, perhaps by actually giving an arrangement of numbers such that in every partial system a first number can be pointed out.

David Hilbert ( 1900 Paris)
 
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as usual, my modus operandi will be to ask about the first thing i don't understand or disagree with and ask that.

Originally posted by phoenixthoth
i agree that complete can't, but why "all?" the quantifier is as in "all sets". why can we not use this? the "therefore" seems to be a non sequitor: the conclusion doesn't follow from the premise.

saying the empty set is {} is meaningless. you have to define it and give it properties such as for all sets x x is not an element of {}.

saying the (absolutely) infinite set {__} without properties is also meaningless. you have to say that for all sets x x is an element of {}. in your theory, you also have to show how russell's paradox is avoided.
 
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Hi phoenixthoth,

To get A as an Empty set we have to define it like that:

x is something

if A is a set such that for any x,x not in A, then A={}(=Empty set) .

But:

x is nothing

if A is a set such that for any x,x not in A, then A=a Non-empty set .

As you can see the property of A depends on the property of x.

For example:

The axiom of the empty(XOR non-empty) set:

if A is a set such that for any x,x not in A, then A=(depends on the property of x, which can be at least something XOR nothing).

As you can see,i used the word 'any' instead of 'all', which give me the possability to deal with both finitely or infinitely many objects.

Because the proprty of GIF ( where GIF is '{' and '}' ) depends on the property of x (as we can show in the above example), no set includes the nagation of its own property(ies).

Thearefore the set of all sets that do not include themselves as members of themselves, does not include itself as a member of itself, because this is its own property, which is: not to include itself as a member of itself.

On a set of infinitely many objects, there is no paradox because we cannot use the word 'all' (or 'complete') with infinitely many objects.

And why we cant use the words 'all' or 'complete' with infinitely many objects?

because:

There are 2 basic types of information that can be examined through GIF.

1.a) Empty set ={}
2.a) Non-empty set

(2.a) has 3 non-empty set’s types:

1.b) Finitely many objects ( {a,b} ).
2.b) Infinitely many objects ( {a,b,…} ).
3.b) Infinite object ( {__} ).

(3.b) Is the opposite of the Empty set, therefore {__}=Full set.

GIF has two limits:

The lowest limit is {}(=Empty set).

The highest limit is {__}(=Full set).

Both limits are unreachable by (1.b) or (2.b) non-empty set’s types.

Or in another words:

Any information system exists in the open interval of ({},{__}).


Organic
 
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The first man whon you meet on the street

Mathematical Problems
International Congress of Mathematicians at Paris in 1900
David Hilbert


Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?
History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future.
The deep significance of certain problems for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.
It is difficult and often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: "A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street."
 
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Hi Moshek,

Let us examine GIF and the CH problem.

GIF is limited by ({},{__})

Where {} or {__} are actual infinity.

We can use concepts like 'order', 'points' and so on, only with finitely XOR infinitely many objects.

Infinitely many objects has the basic property, which is: not to be completed.

Also infinitely many objects cannot be but a potential infinity.

Because R is based on potential infinity it does not have
the power of {__}(=Full-set).

Only {__} can use the model of a line (finite or infinite), therefore no infinitely many points can be a line.

So, any number system that we define, exists in the open interval of
(Empty-set, Full-set).

Now, what is the most simple meaning of the word 'order' when we use it in GIF?

Order is first of all the result of an association between opposite concepts, where in GIF the opposites are ({},{__}).

Actually GIF is only a framework, therefore the more accurate subject of this post, is what I call Complementary Logic (CL) that can be found here:

http://www.geocities.com/complementarytheory/CompLogic.pdf

When we think of 'order' as associations between opposite concepts, this point of view leads us to write things like:

http://www.geocities.com/complementarytheory/ET.pdf

http://www.geocities.com/complementarytheory/CATheory.pdf

These two examples are still non-formal papers, but they can give the general idea of this point of view.


Organic
 
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Mabey Cantor was not so please with Forching.

2. The compatibility of the arithmetical axioms

When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another.
But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results.
In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. Any contradiction in the deductions from the geometrical axioms must thereupon be recognizable in the arithmetic of this field of numbers. In this way the desired proof for the compatibility of the geometrical axioms is made to depend upon the theorem of the compatibility of the arithmetical axioms.
On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms. The axioms of arithmetic are essentially nothing else than the known rules of calculation, with the addition of the axiom of continuity. I recently collected them4 and in so doing replaced the axiom of continuity by two simpler axioms, namely, the well-known axiom of Archimedes, and a new axiom essentially as follows: that numbers form a system of things which is capable of no further extension, as long as all the other axioms hold (axiom of completeness). I am convinced that it must be possible to find a direct proof for the compatibility of the arithmetical axioms, by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers.
To show the significance of the problem from another point of view, I add the following observation: If contradictory attributes be assigned to a concept, I say, that mathematically the concept does not exist. So, for example, a real number whose square is -l does not exist mathematically. But if it can be proved that the attributes assigned to the concept can never lead to a contradiction by the application of a finite number of logical processes, I say that the mathematical existence of the concept (for example, of a number or a function which satisfies certain conditions) is thereby proved. In the case before us, where we are concerned with the axioms of real numbers in arithmetic, the proof of the compatibility of the axioms is at the same time the proof of the mathematical existence of the complete system of real numbers or of the continuum. Indeed, when the proof for the compatibility of the axioms shall be fully accomplished, the doubts which have been expressed occasionally as to the existence of the complete system of real numbers will become totally groundless. The totality of real numbers, i. e., the continuum according to the point of view just indicated, is not the totality of all possible series in decimal fractions, or of all possible laws according to which the elements of a fundamental sequence may proceed. It is rather a system of things whose mutual relations are governed by the axioms set up and for which all propositions, and only those, are true which can be derived from the axioms by a finite number of logical processes. In my opinion, the concept of the continuum is strictly logically tenable in this sense only. It seems to me, indeed, that this corresponds best also to what experience and intuition tell us. The concept of the continuum or even that of the system of all functions exists, then, in exactly the same sense as the system of integral, rational numbers, for example, or as Cantor's higher classes of numbers and cardinal numbers. For I am convinced that the existence of the latter, just as that of the continuum, can be proved in the sense I have described; unlike the system of all cardinal numbers or of all Cantor s alephs, for which, as may be shown, a system of axioms, compatible in my sense, cannot be set up. Either of these systems is, therefore, according to my terminology, mathematically non-existent.
From the field of the foundations of geometry I should like to mention the following problem:


David Hilbert ( 1900 Paris , The second problem)
 
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One of the most meaningful tests that can be used to check some information system, is the test of symmetry.

Let us check N Q and R through the symmetry point of view.


These number systems are divided to "left" and "right" sides by what we call "floating point".

The left is the integer side and right is the fractional side.

Fractional side includes numbers with finitely or infinitely many digits, where the left side includes number with only finitely many digits.

Is there any deep reason for this non-symmetric state between left an right sides?

In my opinion the answer is: no.

And my answer is 'no' because of this reason:


Left and right sides are first of all information cells, which represent our number system upon infinitely many scales, where the rights side is interpolations over scale that approaching to {}, and the left side is extrapolation over scales that approaching {__}.

And by using the word 'approaching' i DO NOT mean become closer to...' but 'cannot reach'.

Therefore from a symmetrical point of view, there are at least two "new" number systems, which are:

Ratural numbers and Irratural numbers.

Ratural numbers are the mirror image of the rational numbers with infinitely many digits, and Irratural numbers are the image mirror of irrational numbers.

An example of this idea can be found here:

http://www.geocities.com/complementarytheory/RiemannsBall.pdf

And more general view of this symmetry can be found here:

http://www.geocities.com/complementarytheory/LIM.pdf


Organic
 
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Organic,

Are you are talking about some new definition of the natural number
that make them possible to be infinite in there magnitude by there structure?

Moshek
 
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Hi Moshek,

I don’t think that Ratural and Irratural numbers are connected to the wall known natural numbers.

From a symmetry point of view we can define this symmetry in both sides of the floating point:


N <--> Q with finitely many digits.

Raturals <--> Q with infinitely many digits (periodic sequence of digits).

Irraturals <--> R (non-periodic sequence of digits).

I used the names Ratural and Irratural to say the these two number systems are in the left side of the floating point, which is the side of the Natural numbers representation.



Organic
 
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Do you mean periodic is necessarily finitely,
in the sense of information?

11001100110011001100... if finity
11010011001101110011... infinty in potential
 
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Periodic is finitely or infinitely many repetitions over scales.

Therefore your both examples can be finitely or infinitely many repetitions over scales.

But again, no one of them is actual infinity ( that can be represented only by {} or (__) ), but a potential infinity that represented by infinitely many objects.
 
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hi Organic,

maybe it is a stupid question of a child,

Please tel me again what is the different between

Actual infinity and potential infinity ?
 
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I read your GIF

Hi Organic,

I just read your GIF new theory,
thank you.

I must tell you that it is not the way mathematician write mathematics, but this is not a problem for me at all.
Sometimes intuition is much more important !

Pleas tell me:

I understood that the full set {__} is actual infinity
But how come that an empty set {}
is also actual infinity.

Moshek
:smile:
 
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To get A as an Empty set we have to define it like that:

x is something

if A is a set such that for any x,x not in A, then A={}(=Empty set) .

But:

x is nothing
the empty set is something. however, it contains nothing. the information it contains is none at all. it is the only set that contains nothing. on the other hand, {__} (which is U in my notation) is something that contains information about all sets. it is the only set with this property.
 
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Hi phoenixthoth,

A set without its content is only a framework.

Its name depends on the property of its content.


By using the ZF axiom of the empty-set, set A name is the opposite of x, where x can be nothing or something.
 
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potential infinity:
{1,2,3,...}
actual infinity:
U or {__}

is that right?

i agree that {} is not an actual infinity. nor is it a potential infinity.
 
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hmm... interesting.
 
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Yes it is very intersting Organic.
 
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The end of Hilbert lecture

...( [The hidden 24 problem] )

The problems mentioned are merely samples of problems, yet they will suffice to show how rich, how manifold and how extensive the mathematical science of today is, and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it. Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly.
But, we ask, with the extension of mathematical knowledge will it not finally become impossible for the single investigator to embrace all departments of this knowledge? In answer let me point out how thoroughly it is ingrained in mathematical science that every real advance goes hand in hand with the invention of sharper tools and simpler methods which at the same time assist in understanding earlier theories and cast aside older more complicated developments. It is therefore possible for the individual investigator, when he makes these sharper tools and simpler methods his own, to find his way more easily in the various branches of mathematics than is possible in any other science.


The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfil this high mission, may the new century bring it gifted masters and many zealous and enthusiastic disciples!


David Hilbert ( 1900 Paris)
 
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Dear moshek and dear phoenixthoth,

I was looking for this beautiful word (interesting) for almost 20 years.

I am so happy, I fill that i am not alone anymore.


Thank you from the bottom of my heart.


Sincerely yours,


Organic
 
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Dear Organic,

You have dedicate 20 years
of your life to create some
new attitude to mathematics
as a whole.

And you are not alone anymore today.

I believe like you that
the creation of "Organic mathematic"
is the solution to Hilbert 24 hidden problem
and also to the 6 th' problem of Hilbert.

And this was Hilbert real vision.

Thank you very much
for sharing us your deep heart.

Your's

Moshek
 

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