Can we prove the correspondence between a real number with a point ?

[fxdung]

Can we prove that there is a corespondence 1-1 between a real number with a point of straight line?It seems to me that we can use Dedekin's cut to prove each point of line corresponds a real number. But how about the reverse?

 

[mathman]

I am not sure, but a first step would be the definition of a point.

 

[fxdung]

How can we definite a point in Euclide geometry?

 

[andrewkirk]

That depends on how you define a straight line. More generally, it depends on what geometry you are using.

Some presentations of what we call Euclidean Geometry do so by effectively defining lines to be images of 1-1 functions from the real numbers, so in that case the answer is, almost trivially, Yes.

On the other hand, I think the original Euclidean geometry, using Euclid's original postulates, only allows construction of items that can be drawn on paper with a straightedge (ruler), pencil and collapsing compass (it collapses when you take either the pencil or the point off the paper). Many constructions cannot be done in such a geometry, such as trisecting an arbitrary angle. While we can construct line segments with square roots as lengths, and maybe even all algebraic numbers, I expect most transcendental numbers are impossible to construct. So we would be unable to establish a 1-1 correspondence between the real numbers and the points on a line in the original Euclidean geometry, because the latter provides no way to identify points whose distance from a given point on the line is a transcendental number.

 

[fxdung]

If we define straight line to be image 1-1 of real number set, so in this geometry the 1-1 correspondence is a axiom, not a theorem?In elementary Euclid geometry,they do not define straight line but they image straight line as light ray. In this case the 1-1 correspondence is a axiom?

 

[andrewkirk]

If we define straight line to be image 1-1 of real number set, so in this geometry the 1-1 correspondence is a axiom, not a theorem?In elementary Euclid geometry,they do not define straight line but they image straight line as light ray. In this case the 1-1 correspondence is a axiom?

Yes to the first one - in that case the correspondence is given as an axiom.

No to the second case, which does not correctly represent Euclid's original postulates. He said nothing about light rays. In fact he does not even define lines or points, simply taking letting them be disjoint categories of unknown fundamental objects. What his axioms (he calls them postulates) do is attribute some properties to lines and points. Those properties are not sufficient to establish a correspondence between points on a line and the real numbers. They may be sufficient to establish a correspondence to some subset of the algebraic numbers, which are a proper subset of the real numbers, but I expect it would take a lot of work to construct one.

How can we definite a point in Euclide geometry?

Typically we take it to be the intersection of two lines. So in Euclid's original geometry we cannot establish the existence of any points that we cannot construct as the intersection of two lines, using only the tools mentioned above.

 

[Office_Shredder]

They may be sufficient to establish a correspondence to some subset of the algebraic numbers, which are a proper subset of the real numbers, but I expect it would take a lot of work to construct one.

Indeed, it's called the set of constructible numbers

https://en.m.wikipedia.org/wiki/Constructible_number

The set was only fully understood (though I think people basically knew what it would contain for much longer, proving it was impossible) in the 1800s with the development of Galois theory.

 

[Ssnow]

You can try by contradiction, assume that "there is a real number that is not in correspondence with a point on a line" ...
Ssnow