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

[Lama]

"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 the real-line scale is fixed, then how a part of the real-line has the same magnitude of the real-line?

 

[Gokul43201]

Two sets have the same cardinality if you can find a one-to-one correspondence that maps one set onto the other.

 

[Hurkyl]

Consider this an advance warning, Organic; if you try to use this thread for theory development, it will be deleted. You've used up all of your second chances.

 

[Lama]

Ok Hurkyl,

I got it, can you please answer to my question?

 

[Gokul43201]

Consider this an advance warning, Organic; if you try to use this thread for theory development, it will be deleted. You've used up all of your second chances.

Organic ? Who's Organic ?

 

[Gokul43201]

Ok Hurkyl,

I got it, can you please answer to my question?

Cantor proved this - try Googling it. In fact $$R^n$$ - n-dimensional real space - is as big too.

 

[Hurkyl]

If the real-line scale is fixed, then how a part of the real-line has the same magnitude of the real-line?

Well, the main question is what do you mean by "part" and what do you mean by "magnitude"?


I will assume by "part" you mean an interval whose endpoints are not the same.

If by "magnitude" you mean simply the cardinality of the sets, then it's done by showing a 1-1 correspondence. For example, a 1-1 correspondence between the interval (0, 1) and the whole real line can be given by the function f(x) = (1-2x)/(x (1-x)); this function maps from the interval (0, 1) onto the entire real line.


If by "magnitude" you mean length, then they generally don't have the same magnitude; the real line has infinite length, while most intervals don't.

 

[Lama]

Hurkyl,

I am talking about a bijection between the real-line and any arbitrary part (or a subset if you will) of it.

So, If the real-line scale is fixed, then how a part of the real-line has the same magnitude of the real-line?

 

[arildno]

What is a "fixed real line scale", and whatever has that to do with the existence of bijections

 

[Hurkyl]

Well, the choice of scale is irrelevant to cardinailty; the line consists of the exact same points no matter which ones I choose to call "0" and "1". So, the question reduces to:

how a part of the real-line has the same magnitude of the real-line?

I gave an example of how the interval (0, 1) has the same cardinality as the real line, by presenting the bijection f(x) = (1 - 2x) / (x (1 - x)).

Here is an example of how a part might not have the same cardinality as the real line: the set of points {0, 1, 2} has finite cardinality, but the real line does not have finite cardinality.

 

[Lama]

the choice of scale is irrelevant to cardinailty

I am sorry, but please look at http://mathworld.wolfram.com/RealLine.html

As you see, the words "fixed scale" are used.

 

[HallsofIvy]

I am sorry, but please look at http://mathworld.wolfram.com/RealLine.html

As you see, the words "fixed scale" are used.

He didn't say you should use the words, he said they were not irrelevant to cardinality.

 

[matt grime]

Organic, www, lama, dialog, whatever today, these things are all definitions. Why do you not accept that? Fixed scale is not a well defined term, and I would hesitate to say what it means, but presumably it is to do with distances being euclidean. That has nothing to do with cardinality. "size" is a vague word, and just as we would never say "amount" when referring to chemistry if we were being precise, you need to distinguish between the suggestive descriptions offered as an aid to understanding and the formal rigorous definitions that are used when we come to practise mathematics.

 

[Shemesh]

I have found this thread https://www.physicsforums.com/showthread.php?p=228438#post228438 .

 

[matt grime]

Please can someone lock this? Don't suppose he's on a static IP is he?

 

[Shemesh]

Can you explain f(x) = (1 - 2x) / (x (1 - x))?

 

[HallsofIvy]

What do you mean "explain" it? It's a rational function defined for all real numbers except and 1. It has vertical asymptotes at x= 0 and x= 1 and a horizontal asymptote y= 0. What more do you want to know about it?

 

[Shemesh]

This function maps from the interval (0, 1) onto the entire real line.

But it is not explain how it can be done if the real line has no fractal structure (self similarity over scales).

So, where can I find a mathematical theory that explains why it can be done?

 

[matt grime]

'it' needs to be defined before we can explain 'it'

 

[Shemesh]

Please give this definition.

 

[matt grime]

erm, it's your 'it' that you need to define. because your post appears to say that there can be no bijection from (0,1) to R because we don't take into account the 'fractal' nature of it, which is obviously garbage. so what then is the 'it' we need to explain to you?

 

[Shemesh]

...which is obviously garbage...

By what knowledge do you come to that conclusion?

Pease share it with us.

 

[matt grime]

because without needing to define 'fractal' the function tan(x) takes the interval (-pi/2,pi/2) bijectively onto R. that is why we can conclude constructively that it was garbage. That is in addition to the example you failed to understand when posted by HallsofIvy earlier in the thread. Do you ever read and understand the response people bother to give?

 

[Shemesh]

If we can define the same magnutide among infinitaly many different intervals why can't we conclude that there is a self-similarity between these intervals?

 

[master_coda]

If we can define the same magnutide among infinitaly many different intervals why can't we conclude that there is a self-similarity between these intervals?

Because the fact that you can find a bijection between different intervals only proves that you can find a bijection between different intervals. You can't just assume that the intervals must also have other similarities if they have one.

 

[matt grime]

And as proof of that, the Koch snowflake of fractal dimension log2/log3, I think, is in bijection with the real line. The unit sqaure is in bijection with the unit interval as is the Cantor set. So, erm, there you go, again.

 

[Shemesh]

[

You can't just assume that the intervals must also have other similarities if they have one.

Please show me where I say that there is more then one self similarity over-scales to the real-line?

A one self-similarity over scales, this is exactly the definition of a fractal (and Cantor's set http://mathworld.wolfram.com/CantorSet.html proves it Matt).

In short the real-line is a fractal because any unique "fixed place" R member is also used as a global scale factor on the entire real line, as we can see here:

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


Why are you all against this idea?

 

[matt grime]

eh? what has that to do with the disproof of your assertion that because there is a bijection there must be a fractal nature to it and self similarity over scales, whatever that might mean? bijections have no properties that imply they preserve fractality, for want of a better word. indeed the counter examples i gave you seem to ignore, again, why do you always do that? The cantor set and the real line are in bijective correspondence, as is the power set of the integers and the koch snowflake, and there is no fractal link there between those things is there? they don't all have the same fractal dimension, one isn't even a fractal.

 

[matt grime]

Why are you all against this idea?

because it's ill thought out, poorly presented, badly explained, uses mathematics wrongly, indeed it is wrong to say that the existence of a bijection must be to do with the fractal nature of the real line, and because, quite frankly, it's you, and your reputation precedes you.

no one would argue against R being a fractal, with fractal dimension 1. People will, have, argued that because R and (0,1) are in bijective correspsondence does not tell you anything about the self similarity of (0,1). the real line and the parabola y=x^2 are in bijective correspondence, and the parabola is not self similar.

 

[Shemesh]

I am not talking about some spesific fractal dimension of the real-line, but on the general principle that the real-line is THE "ULTIMATIVE" FRACTAL of any existing fractal including fractal dimension 1.