Can a Segment of the Real-Line Have the Same Magnitude as the Whole Real-Line?

[Shemesh]

An explanation of Vacuous Truth can be found here: http://en.wikipedia.org/wiki/Vacuously_true#Vacuous_truths_in_mathematics

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 global scale factor on the entire real-line, which its product is the entire real-line included in itself according to this global 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.

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.


Any comments?

 

[matt grime]

if we take just positive real numbers, the same is true of the right half of the parabola y=x*x, as it is of the tan curve in its principal region. neither is a fractal. or have you just redefind fractal to suit your purposes?

 

[Shemesh]

You do not understand my argument.

I am not talking about the proprty of some function but on the way of how the magnitude of its elements can be the same in any sub-collection of it.

The fractal nature of the magnitude of R collection do not care about the "character" of any function that has the magnitude of R collection.

For example, take any segment of your non-fractal parabola, and the magnitude remains the same.

By using the word "fractal" I mean self-similarity of the magnitude that can be found in any sub-collection of your function.

 

[matt grime]

so fractals have nothing to do with it at all? why am i not even moderately surprised?

 

[Shemesh]

Matt, self-similarity of some property (and in this case the magnitude of R collection) which can be found in any arbitrary sub-part of the examined system (and in this case the system is the real-line) is nothing but a fractal, got it?

 

[matt grime]

so something is a fractal in your new sense if there is a bijection to some proper subset of it? that would be dedekind infinite then, wouldn't it?

 

[Shemesh]

bijection to some proper subset of it is the clearest sign that we have here a fractal.

I think Cantor also used this property.

 

[matt grime]

no, it is just dedekind infinite. it doesn't seem reasonable to say it is therefore a fractal, unless you want to redefine the word fractal to mean dedekind infinite, which is what you're are saying

 

[Shemesh]

Dedekind used this property, but he did not know that it is actually the property of a fractal.

Cantor defined the Cantor set, but he did not know that it is a fractal.

Chaos theory was developed after their time.


(Also please pay attantion that our standard place value representation method is also a fractal and also surreal numbers)

 

[matt grime]

"it is actually the property of a fractal" oh, that makes it all so much clearer...

 

[Shemesh]

When a collection of elements refers to itself, you can get a fractal.

 

[hello3719]

Shemesh, where is that all going? So what if the real line can be represented in different scales.It doesn't mean that you will find "new" properties. If a property of fractals appears with some representations doesn't mean that the real line will have all of the properties of fractals.

 

[master_coda]

This isn't going anywhere. This is just an attempt to change to meaning of the word fractal to something general and vague for no reason.

If someone was interested in trying to describe a new or interesting idea, they would use the existing terms with their existing definitions instead wasting all their time redefining things so that nobody can understand what they're talking about.