What Are the Properties of a Point and a Line?

[Didd]

Hello everybody,

Between two points on a line there are infinitely many points.

Reason: A point have no dimension(no width,length or height)

Do you have another idea ?

 

[MathematicalPhysicist]

that's seems to be more philosophical question.
and to spice things, let's say we have one line which is equal 1cm and another which equals 2 cm in which line there will be more points?

 

[MathematicalPhysicist]

and by the way both the line and a point are undefined concepts in geometry.

so you can also say in one line there are infinite many lines.

 

[Didd]

You have a line. Plot a segment. And the segments have two end points.Then between this points there are infinitly many points. Not only that but between any two points on line there are infinitly many points.

That is what I am saying.

 

[Didd]

And I am verry sorry for not making my idea clear. I just wanted if some one have another reason on the topic.

 

[rayjohn01]

Didd -- if you only consider this from a maths viewpoint what you say is the case there are infinitely many points even if the line is as short as you can imagine. Being an engineer I prefer to avoid this by thinking more of natures lines which when we measure them are made up of physical objects ( usually a finite number) -- not just that but they are moving so precision is limited anyway.

 

[Gokul43201]

I think what you are trying to say is that there are an infinite number of real numbers in any real interval. The argument you provide is not a rigorous mathematic proof. To say that "a point has no dimension" has little meaning in math.

A better way to approach the question is by understanding the cardinality of such a set, and seeing why it is not finite.

 

[Hurkyl]

It's a fun exercise to prove from the axioms of Euclidean Geometry that, given two points on a line, there exists a point between them.

Then, given this new point, you pair it with one of the original points to get another one, and so on.

Once you can prove that each new point "discovered" is different from all the previous ones, this yields an inductive argument that the number of points between the original two points cannot be finite.


PS. IIRC, Euclid's 5 axioms are inadequate, you have to use a more modern formulation.

 

[zeronem]

This is actually the same concept of the Banach-Tarski Paradox. Whereas they take pieces out of the Mathematical Sphere and from those pieces form another sphere of the exact same size as the original sphere through rotation and translation of the pieces. This is due to that the 3 Dimensional Unit sphere is made up of infinitely many points I believe. However The same concept you are talking about is the main concept in the Banach-Tarski Paradox. Correct me if I'm wrong.

 

[fourier jr]

This is actually the same concept of the Banach-Tarski Paradox. Whereas they take pieces out of the Mathematical Sphere and from those pieces form another sphere of the exact same size as the original sphere through rotation and translation of the pieces. This is due to that the 3 Dimensional Unit sphere is made up of infinitely many points I believe. However The same concept you are talking about is the main concept in the Banach-Tarski Paradox. Correct me if I'm wrong.

The Banach-Tarski paradox is one thing that happens if you allow the Axiom of Choice. Not sure hoe that works because the textbook I looked at (the one by Suppes) only mentioned it in passing. I think the Banach-Tarski paradox is probably one of the reasons that people don't want to use the Axiom of Choice or its equivalents.

 

[zeronem]

So would Didd be applying the Axiom of Choice with his idea?

Lets say each point is a set of points. And those set of points are also a set of points and so on and so on. Now I am getting into Infinitie sets. Could this be a similar approximation to the line with 2 points plotted at a distance, with infinitly many points in between. We get a structure of points everywhere with this idea of sets and points. If we add even more precision, could we not form a 3 dimensional Lattice? Being lattice points making up a 3 dimensional space. Each Lattice point containing a set of Lattice points.

The more we dive in, the larger the 3 dimensional Space gets. So we could say that a 3 dimensional space with a particular boundarie not being Infinite is mapped by infinite amount of Lattice points. The set of Lattice points of a Lattice point is also contained within the 3 dimensional space. Depending on what degree of set you are in of the lattice points depend on how big the 3 dimensional space of finite boundary is. Since the 3 dimensional lattice points are infinite within the 3 dimensional space, we can take pieces of this 3 dimensional space and form a 3 dimensional space of the same size as the orginal 3 dimensional space with the pieces.

I did form from this idea of his, a paradox extremely similar to Banach-Tarski Paradox. In fact, just place the 3 dimensional space with finite boundary, made up of infinite lattice points with a 3 dimensionsal sphere with finite boundary. Even though Didd, did not state if the line was infinite or not, he did describe a section of a line being that he brought in the concept of points plotted on the line. Particularly the two points he mentioned. Either his line is of finite boundary or infinite boundary, the concept still developed that, there are infinite many points in between two points.

 

[Didd]

The point I am speaking about is the geometric point. The statement is given as an assumption.When I see it, the reason for the statement is as I stated before.
You know maths is mainly an assumption(or agreement). If we agree that 1+1=3, what's the problem? It is just an agreement.

Note: But an agrement is not provable. It is just an agreement. However, I am giving my idea of how they agreed on this statement. Going to prove an agreement... . Don't think about it.

 

[MathematicalPhysicist]

another thing, didd, i would recommend you to read in the web about cantor's comb.
it makes the similarities mentioned by gokul between a line and the real numbers.

 

[Didd]

That is the reason why we assoisiated real numbers with points on a line. It is from this statements "In a line there are infinitly many points" and " Between a two points on line there are infinitly many points."

 

[Hurkyl]

The problem with dropping the axiom of choice is that you get other annoying paradoxes. For example, the "sock paradox":

I have an infinite number of drawers, and each drawer contains two pairs of socks; the red pair of socks has a distinct left and right sock, but the white pair of socks has two identical socks.

There is a choice function that let's me select a red sock from each drawer. (e.g. pick the left sock) However, without the axiom of choice, there does not exist a choice function that let's me select a white sock from each drawer.



Anyways, it's somewhat silly to drop the axiom of choice because of the Banach-Tarski paradox; the pieces involved are so strange that the concept of volume cannot apply to them. Thus, it shouldn't be any surprise that manipulating these things can disobey the "laws" obeyed by things that do have volume.

 

[matt grime]

organic,www,shemesh, etc strikes again?

Hurkyl is absolutely correct to point out that the segments in Banach Tarski are not measurable (in the correct sense which you'll need to look up Doron rather than presume you know it already; apologies if it isn't you)



"That is the reason why we assoisiated real numbers with points on a line. It is from this statements "In a line there are infinitly many points" and " Between a two points on line there are infinitly many points.""

that is false since the same applies to the rationals.

they can be thought of as a line since they are the completion of Q and are ordered.

 

[ExecNight]

I just got this logical conclusion at this very moment..


1 : A point has no dimensions.

2 : Everything that exists in universe has dimensions.

3 : Then a point can only be described as noting in our universe.

4 : Noting is described by the number Zero ( 0 ) in mathematics.

5 : To find the number of points in a line you must divide it by the value of a point.



Conclusion : A value can not be divided by Zero ( 0 ).So the statement that there are infinite points in a line is invalid.

 

[zeronem]

I just got this logical conclusion at this very moment..




2 : Everything that exists in universe has dimensions.

Conclusion : A value can not be divided by Zero ( 0 ).So the statement that there are infinite points in a line is invalid.

Number 2 is the problem. You brought in the idea that everything in the Universe has dimensions. We are not exactly talking about the Universe. Your statement could probably be valid in Physics. However in Mathematics, we come across symbolic logic, whereas we didn't use the "Universe" as a symbol in this area of Logic we are discussing.

 

[ExecNight]

Replace the universe with Space-Time Coordinate System.

Looks valid? If not please be more clear so i may be able to answer your question.

 

[zeronem]

Space and time is not exactly what we are talking about. Didd was simply introducing a logical concept absent of Universal features. It has a lot to do with just understanding the concept that has arised.

If you ever notice, what are algebraic equations? What do they mean? Algebraic Equations have nothing to do with the Universe, until it is applied to Physics, or chemistry. However the plain old Algebraic Equation in Mathematics describes nothing. We just work with the properties of Algebra to understand the algebraic equations. It is only to further our development in Mathematical Understanding. Don't get me wrong, we can graph an Algebraic equation in 3-dimensional and 2-dimensional space, however the point I am making is that it is just an idea.

Anyways, Nothing about Space or time is really needed on this concept. This concept can simply be described by Set Theory which is closely associated with Logic.

 

[ExecNight]

Oh if you are getting at that, then you can place infinite lines between two points.So the infinite point in a line between two points statement loses its relevance,while you do not know which line you are talking about between that two points.

Well i am no mathematician.But if you accept that there can be infinite points inside a line,but there can't be infinite lines between two points then i am sorry to say that algebraic equations are nonsense for me.

I am just using logic here

 

[zeronem]

Now your just switching two things with different meanings to even make it inconsistent and nonsensical. Like I said, we had to study the properties of algebra to understand algebraic equations. Just as we have to study the properties of a point and a line to understand which goes which. So if you look at the Didd's definition of a point as well as a point on a line, being metaphorical to a real number on a line. We take the properties of mathematics to understand exactly the idea he is trying to get across. Hence, mathematical axioms are what we use to further understand the idea he has posted on here. Set Theory being closely associated with Logic, established some of the known axioms in Mathematics.