The Real-Line has a fractal nature

[Lama]

The Real-Line also has a fractal nature

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The standard definition:

"Real line" is used to mean real axis, i.e., a line with a fixed scale so that every real number corresponds to a unique point on the line. (http://mathworld.wolfram.com/RealLine.html)
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If you look at http://www.geocities.com/complementarytheory/Real-Line.pdf , you can see that by this model any member (or element) of R set can be simultaneously in both states:

1) As some unique number of the Real-Line (a unique member of R set)

2) As a scale factor on the entire Real-Line, which its product is the entire real-line included in itself according to this scale.

There is no process here but a simultaneous existence of R set on infinitely many unique scale levels of itself.

Because of this self-similarity over scales, we can understand why some segment of the Real-Line can have the magnitude of the entire Real-Line.

Please understand that we are not talking about some shape of a fractal, but on the infinitely many levels of non-empty elements, which are included in R set, and they have the magnitude of the entire Real-Line.

It is important to stress that there is one and only one magnitude to the real line, which is not affected by its fractal nature.


Dedekind's Cut:

A set partition of the rational numbers into two nonempty subsets S1 and S2 such that all members of S1 are less than those of S2 and such that S1 has no greatest member. ( http://mathworld.wolfram.com/DedekindCut.html )

If we examine the open right side of S1 we can clearly see that there are non-linear intervals among S1 elements when they tend to S2 first element.

This non-linearity can be found in infinitely many levels of the non-linearity state itself, because of the fractal nature of the Real-Line.


Conclusions:

There is no intermediate state between emptiness {} and non-emptiness {X}.

The minimal non-empty element is a point {.}, which is the first basic state of self-similarity of a non-empty element (only '=' can be used).

Any number can be referred to this point.

The second basic state of a non-empty element is a line segment where its left edge is the first point (with some arbitrary number referred to it) and its right edge is the second point ('=','<','>' can be used).

There is no intermediate state between a state of a point {.} and a state of a line segment {_}.

Any arbitrary number, which is bigger than the first number, can be referred to the second (right) point.

If both arbitrary numbers have been given, then and only then, we can define the entire real numbers in and out of the domain of the first and the second arbitrary numbers.

The proportion among the real numbers is invariant, but it can be found on infinitely many levels of scales of the Real-Line, which are determined by each number of the Real-Line, when it is used as a global scale factor of the entire Real-Line.




Any comments?

 

[Lama]

Any proper sub-interval will have to have at least a greatest lower bound or a least upper bound.

Why do you come to this conclusion?

The internal proportion of each level remains the same, no matter what scale factor we are using.

It means that actually there is no such a thing like "sub-interval" from the internal point of view of each level.

You said an element can be in the second state. An element of the R set is analogous to a point on the Real Line, not a segment of the Real Line or the whole Line itself. A point is a zero-dimensional object, a line is a 1-dimensional object. To say that one can be seen as analogous to the other doesn't work.

Why do you come to the conclusion that the second state is related to a line?

for example: pi is an element of the real line and have a 1-1 map with some point on the real line.

But pi can also be a scale factor on the entire real line (what I call a second state):

Let x1 be any arbitrary real number, which is used as a scale factor on the entire real line.
Let x2 be any arbitrary real number.
Let x3 be the result of the equation x1*x2=x3.

If x1 = pi then we can get the sequence:

x3 = pi*x2 where x2= 4,
x3 = pi*x2 where x2=pi,
x3 = pi*x2 where x2=e,
x3 = pi*x2 where x2=0,
x3 = pi*x2 where x2=-3.456764...,
x3 = pi*x2 where x2=.000043267,

... and so on.


In general, any arbitrary real number x2 can be used as the value of x1 of the equation x1*x2=x3.

Now, because multiplication is commutative in standard real analysis, then x3 results are actually the entire numbers of the real line, which means that x3 results are always the same, no matter what x1 scale factor is used.

Strictly speaking, the internal proportion of intervals between the real numbers remains the same, but this internal proportion can be found on infinitely many different and unique scale levels which are determined
by the entire real numbers, when they are used as x1 values.

In short, the real-line has a fractal nature.

 

[Lama]

Any comments to the first and the second posts?

Maybe this analogy can help to understand my idea:

Let us say that we have two universes A and B, which are exactly the same in any detail of them, but A is twice bigger than B.

It means that if we refer unique names (numbers) to each fixed place (some point) on the real-line, and then both real lines of each universe will be compared to each other, by using unique name (number) '0' as a common point of both systems, then we shell find this state:

Universe A real-line unique names:

0        1/2        1
|_________|_________|

Universe B real-line unique names:

0   1/2   1
|____|____|

So as you see, in this case each universe (0 is excluded) has a different and unique sequence of names (numbers), which is actually related to the same system of points.

Now let us merge universes A and B to a one Mata-Universe MU (A=2B in MU), but also in this Mata-Universe MU the name (number) of each point is determinate, for example, by a point named '1'.

It means that even if both A and B worlds of MU starting from the same point named '0', we get two different systems of 1-1 maps between the same points but with different unique names (numbers) related to them.

Now think about MU with 2^aleph0 worlds like A and B, and you get the picture of my idea.

 

[Lama]

In the sense that you understand it (which is not really the proper "fractal" interpretation of mathematics) you are correct. Any interval of the real line, e.g. [0,1] is isomorphic to R.

This is nothing new however, it is on page 12 of most first year bachelor math/physics books.

But then we can define another interpretation to the Epsilon-Delta argument which is based an included-middle logical system.

Please see my paper in http://us.share.geocities.com/complementarytheory/ME.pdf

Thank you.

 

[Lama]

No point on the real-line is affected by any scale factor, because it has a zero dimension.

Because of this property, a very interesting situation can be observed:

Each point on the real-line actually can have the entire names (numbers) of the real-line.

It means that we get a matrix of 2^aleph0*2^aleph0 names "above" the points.

My first examination of this "cloud of information" can be found here:

http://us.share.geocities.com/complementarytheory/ONN.pdf

 

[Lama]

Hi BigBlueHead,

First, thank you for reading my paper and for your detailed reply.

As for your last question, my point of view is based on what I call Complementary Logic (CL).

Boolean Logic and Fuzzy logic are proper sub-systems of it.

CL can be found in http://us.share.geocities.com/complementarytheory/ME.pdf

CL point of view on Math, is based on the information concept, which includes our cognition ability to use it as an internal (abstract) and external (physical) concept.

In short CL is based on an included-middle reasoning, where two opposites are simultaneously preventing/defining each other, and the result is a middle(=included-middle reasoning).

You cannot understand my work if you use an excluded-middle reasoning, where two opposites are simultaneously contradicting each other, and the result is no-middle(=excluded-middle reasoning).

CL clearly distinguishes between the actual and the potential.

For example:

Because Math is first of all a form of language, CL starts by defining the weak limit (=Emptiness) and the strong limit (=Fullness) which are the non-reachable limits of any information system, including Math.

In short, actual Emptiness (which is represented as the content of {}= the empty-set) and actual Fullness (which is represented as the content of {__}= the full-set) are too weak or too strong to be used as available inputs for Math language, but they determine the lowest and the highest limits of it.

So the only available inputs can be found by finite or infinitely many elements that can be represented by two basic information forms which are {.}=point(s) and/or {._.}=line segment(s).


By CL:

1) There is no intermediate state between emptiness state {} and non-emptiness state {X}.

2) There is no intermediate state between fullness state {__} and {.} or {._.} states.

3) There is no intermediate state between a state of a point {.} and a state of a line segment {._.}.

By {} <--x(={.}) we mean that {.} is a potential {}.

By x(={._.})--> {__}} we mean that {._.} is a potential {__}.

By CL, the least useful input cannot be but a combination of {.} AND {._.} forms, therefore x is at least both {.} AND {._.} information form.

I hope that now you can understand ({},{__}):={x|{} <--x(={.}) AND x(={._.})--> {__}}

Also please look at http://us.share.geocities.com/complementarytheory/ONN.pdf which is my first research of what I call "Organic Natural Numbers" which are based on CL.

 

[Lama]

I have another analogy.

Please take an infinitely long line and use any abitrary positive scale factor > 0 in it (by the way, in scale factor 0 the infinitely long line is actually a point).

In any scale, a line has a self-similarity to itself, which means that a line is actually a fractal.

Now please take this analogy and use it on the real-line.

 

[moshek]

To your definition every real number have local and global property
This impotrant view can be unify to any concept in mathematics by the nature of "Organic mathematics".


Moshek

 

[Lama]

I understand what you're saying, I think. Basically, if you stretch or shrink the real line, you still have the real line. Like if you zoom in on one of those fractal images, you keep getting the same image. Is that your point? What type of applications or implications do you think this might have? However, I'm not too sure, but I think a fractal would need a non-integer dimension, the Real line has an integer dimension: 1.

No point on the real-line is affected by any scale factor, because it has a zero dimension.

Because of this property, a very interesting situation can be observed:

Each point on the real-line actually can have the entire names (numbers) of the real-line.

It means that we get a matrix of 2^aleph0*2^aleph0 names "above" the points.

My first examination of this "cloud of information" can be found here:

http://us.share.geocities.com/complementarytheory/ONN.pdf

the Real line has an integer dimension: 1.

Now we come to the heart of my agument, which is:

The real line is a 1 dim. shadow of a multi-dimansional fractal of infinitely many information forms in infinitely many symmetrical-levels and clarity degrees.

 

[Lama]

To your definition every real number have local and global property
This impotrant view can be unify to any concept in mathematics by the nature of "Organic mathematics".

This is beautiful dear Moshek. Thank you very very much.

By the way, from this global/local property of each real number we can learn that the real line is both relative and absolute system because of a very beautiful and simple reason, which is:

The real line is based on two building-blocks: a point {.} and a line segment {._.}.

The line segment, which is a 1 dimension element, is affected by the scale factor that is operated by some real number on the entire real-line. In this case the real-line is changed relatively to the scale factor.

The point, which is a 0 dimension element, is not affected by any scale factor, and we get an absolute system.

As a result, each absolute point on the real-line, can have the names (numbers) of the entire real line and we get a matrix of 2^aleph0*2^aleph0 names (numbers) "above" the real-line.

My first research of the finite n^n matrix can be found here: http://us.share.geocities.com/complementarytheory/ONN.pdf

From this point of view the language of Mathematics is the "story" of the interaction between the relative and the absolute concepts.

 

[hello3719]

Are you enjoying talking with yourself ?

 

[Lama]

Dear hello3719,

I have a multi-dialog on this subject on several forums, and I copeid some of them to this thread.

I'll be glad to get also your remarks and insights.

Thank you.

 

[Lama]

Please read posts #1 - #10 before you air your view about this thread.