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Proving countable ordinal embeds in R

  1. Feb 2, 2016 #1
    1. The problem statement, all variables and given/known data
    show that if q is any countable ordinal, then there is a countable set A ⊆ R (in fact we can require A ⊆ Q), so that (A, <) ∼= (q, ∈).
    3. The attempt at a solution
    since q is a countable ordinal this implies that it has a mapping to the naturals.
    to me this seems strong enough. and its also well ordered.
     
  2. jcsd
  3. Feb 2, 2016 #2

    andrewkirk

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    It's hard to tell from the notation you've used, but I suspect the question is asking you to prove there is an order-preserving mapping.
    From countability of q you can get a bijection to the naturals, but in most cases it won't be order preserving.
     
    Last edited: Feb 4, 2016
  4. Feb 3, 2016 #3
    in order to preserve order, can we use a permutation group to create a linear order
     
  5. Feb 3, 2016 #4

    andrewkirk

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    The permutation groups I am familiar with are on finite sets. Unless q is finite - in which case the entire problem becomes trivial - I can't see how a permutation on a finite number of elements would help. I think you will need to use the structure of countable ordinals to prove this.
     
  6. Feb 4, 2016 #5

    micromass

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    Use transfinite induction and the fact that ##\mathbb{R}## is order isomorphic to any interval ##(a,b)##.
     
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