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Dedekind cuts

  1. Jan 4, 2014 #1
    If every Dedekind cut is at a rational it seems that these cuts would only produce a countable set and would not produce the whole real line. So how should I think about it.
     
  2. jcsd
  3. Jan 4, 2014 #2

    mathman

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    The point is that there are Dedekind cuts that are NOT at rationals.
    For example, take any real number A and consider its decimal expansion. A sequence of rational numbers An can be defined by taking n terms of the expansion. Let the cut be defined by all rationals less than any term in the sequence. This cut gives the real number A.
     
    Last edited: Jan 4, 2014
  4. Jan 5, 2014 #3
    Even though all Dedekind cuts consist of only rational numbers, all are not rational cuts.
    T = { x [itex]\in[/itex] Q: x^2 < 2 or x < 0 } is a dedekind cut, you can check that all the properties hold, but it can not be a rational cut.
     
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