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Homework Help: Dedekind infinite sets

  1. Jan 6, 2013 #1
    1. The problem statement, all variables and given/known data
    Call a set X Dedekind infinite if there is a 1-to-1 mapping of X onto
    its proper subset.
    Prove that every countable set is Dedekind infinite.
    3. The attempt at a solution
    I want to say that every countable set can be well ordered.
    I guess I could just pick some element from our set X and call it a.
    And then make sure everything from our set gets mapped to something
    larger than a. So we have a 1-to-1 mapping to our proper subset.
    I probably need to be more rigorous about how this mapping takes place.
  2. jcsd
  3. Jan 6, 2013 #2


    User Avatar
    Science Advisor
    Homework Helper

    Yes, you should be more explicit about what the mapping is. What's the definition of countable set?
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