The Real-Line is a fractal

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"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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There are two basic states that stand in the basis of the real-line, which are:

a) = (self identity).

b) < or > (no self identity).


Let x be a real number.

Any real number, which is not x cannot be but < or > than x.

The difference between x and not_x, defines a collection of infinitely many unique real numbers.

The magnitude of this collection can be the same in any sub collection of it, which means that we have a structure of a fractal to the collection of the real numbers.

In short, each real number exists in at least two states:

a) As a member of R (local state).

b) As an operator that defines the fractal level of R (a global operator on R).

Any fractal has two basic properties, absolute and relative.


The absolute property:

Can be defined in any arbitrary level of the fractal, where within the level each real number has its unique "place" on the "real-line".


The relative property:

Any “sub R collection” in this case is actually R collection scaled by some R member as its global operator, and this is exactly the reason why some "sub R collections" can have the same magnitude as R collection.


We can understand it better by this picture:

http://www.geocities.com/complementarytheory/Real-Line.pdf


In short, R collection has properties of a fractal.


What do you think?
 
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