## R set magnitude

"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?
 Recognitions: Gold Member Science Advisor Staff Emeritus Two sets have the same cardinality if you can find a one-to-one correspondence that maps one set onto the other.
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## R set magnitude

Ok Hurkyl,

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

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 Quote by 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.
Organic ? Who's Organic ?

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 Quote by Lama 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.

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 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.
 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?
 Recognitions: Gold Member Homework Help Science Advisor What is a "fixed real line scale", and whatever has that to do with the existence of bijections

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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.

 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.

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