Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Question about a measure of a set

  1. Dec 31, 2012 #1
    Could we use the fact that all countable sets have zero measure
    to prove that their must be a larger infinity.
    we know countable sets have measure zero because I could just start by making
    boxes around each number and then add up their widths.
    For the first number I will make a box that has width [itex] \frac{\epsilon}{2} [/itex]
    and then each box will have half the width of the previous box.
    so the sum will be [itex] \epsilon(1/2+1/4+1/8.........) [/itex]
    and i can make [itex] \epsilon [/itex] as small as I want.
    This proof comes from Gregory Chaitin.
    If the reals were countable they would have measure zero, but we know this isn't true
    because the reals have positive width. Can i do this to prove there is a larger infninty than countable.
     
  2. jcsd
  3. Dec 31, 2012 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Short answer - yes. The only objection, compared to Cantor proof, is that it is necessary to develop measure theory first.
     
  4. Jan 1, 2013 #3
    ok thanks for your answer. Instead of using measure theory could I just talk about lengths and use convergence of this infinite series.
     
  5. Jan 2, 2013 #4

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Yes - although if you look at it closely you will find you are using some elementary facts from measure theory.
     
  6. Jan 2, 2013 #5
    ok thanks
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Question about a measure of a set
Loading...