# Real-line

1. Jun 7, 2004

### Lama

-------------------------------------------------------------------------------------------------
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)
-------------------------------------------------------------------------------------------------

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.