Infinite set of points on a line.

Click For Summary

Discussion Overview

The discussion revolves around the concept of dividing a line segment into infinitely smaller parts and the implications of this process on the nature of points within the segment. Participants explore the meaning of "to infinity," the intersection of line segments, and the relationship between infinite sets of points and limits.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions whether dividing a line segment of size L into halves infinitely results in a point made up of an infinite number of points or if it leads to an indeterminate form of ∞/∞.
  • Another participant seeks clarification on the meaning of "to infinity," suggesting that without understanding this concept, the original question lacks coherence.
  • A participant proposes a method of defining the process through the intersection of line segments, illustrating with a specific example of continually halving the segment from 0 to 1.
  • There is a discussion about whether a line, being composed of infinitely many points, retains that property when halved, with some asserting that half of the line remains a line with infinitely many points.
  • Another participant introduces the idea of taking a limit as n approaches infinity, questioning how this concept applies to sets rather than functions or numbers.

Areas of Agreement / Disagreement

Participants express differing views on the implications of dividing a line segment infinitely and the meaning of "to infinity." There is no consensus on how these concepts should be interpreted or applied.

Contextual Notes

Participants highlight the need for clarity regarding the definition of limits in the context of sets, as well as the implications of removing points through infinite division.

redrum419_7
Messages
58
Reaction score
0
If a line segment of size L, is made up of an infinite amount of points. Then divided into half, and then half again, divided into half to infinity, would this mean that the resulting line segment is a point made up of an infinite amount of points?

Or would it be ∞/∞?
 
Last edited:
Physics news on Phys.org
redrum419_7 said:
Then divided into half, and then half again, divided into half to infinity

What does this mean?? What does "to infinity" mean??
 
What does it mean to do a thing to infinity? Until you are clear on that your question makes no sense.

There is an accepted way to define what you are trying to express which is the intersection of all the line segments.

Let us consider the case where we start with the line segment from 0 to 1 and always choose the left half when we divide. So,
Let L0 be the line segment from 0 to 1.
Let L1 be the line segment from 0 to 1/2.
Let L2 be the line segment from 0 to 1/4.
...
Let Ln be the line segment from 0 to 1/2^n
...


Now let L be the set of points obtained by doing this infinitely many times whatever that means. Clearly whenever we divide we can only remove points, never add. Thus if a point is in L, then it must also be in
L1,L2,L3,...
since L is contained in all of these.

So what points are contained in
L1,L2,...?
Is -1 in all these? Is 0? Is 1/2? Is 1? Is 2?
Once you answer this question I think you should have an idea of how to answer your question in most simple cases.

BTW infinity/infinity doesn't really make any sense in this context, or in most mathematical context for that matter.
 
Line is made up an infinitely amount of points right? Then wouldn't half of that line, still being a line, be made up of an infinitely amount of points?
 
micromass said:
What does this mean?? What does "to infinity" mean??

Taking a limit as n-->∞
 
redrum419_7 said:
Line is made up an infinitely amount of points right? Then wouldn't half of that line, still being a line, be made up of an infinitely amount of points?

Yes.

redrum419_7 said:
Taking a limit as n-->∞

And what does that mean?? Limits is usually defined for functions or numbers. You're working with sets. So what do limits mean?
 

Similar threads

High School Geometric Issues with a line, a plane and a sphere...
  • · Replies 10 ·
Replies
10
Views
2K
High School Axes on a hyperbolic plane
  • · Replies 4 ·
Replies
4
Views
2K
High School East-West error at regular points on the Azimuthal Equidistant Map
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad A set of numbers as a smooth curved changing manifold.
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Maximum separation on an infinite line
  • · Replies 29 ·
Replies
29
Views
3K
Undergrad On a proof of the converse of the supporting hyperplane theorem
  • · Replies 9 ·
Replies
9
Views
3K
Graduate Connecting N points pairwise in volume V, average density of lines?
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Working out a point / segment on a sphere
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Can the area be understood as the "number of points"?
  • · Replies 27 ·
Replies
27
Views
6K
Undergrad About the maximal extension of local charts on a manifold
  • · Replies 8 ·
Replies
8
Views
2K