Infinite set of points on a line.

  1. 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: Mar 16, 2012
  2. jcsd
  3. micromass

    micromass 18,962
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    What does this mean?? What does "to infinity" mean??
     
  4. 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.
     
  5. 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?
     
  6. Taking a limit as n-->∞
     
  7. micromass

    micromass 18,962
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    Yes.

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