"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?
Two sets have the same cardinality if you can find a one-to-one correspondence that maps one set onto the other.
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.
Cantor proved this - try Googling it. In fact [tex]R^n[/tex] - n-dimensional real space - is as big too.
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?
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: 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.
I am sorry, but please look at http://mathworld.wolfram.com/RealLine.html As you see, the words "fixed scale" are used.
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.
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?
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?