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Math Help - Ordering of a set

  1. #1
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    Ordering of a set

    Find a (nonstandard) ordering of the set of natural numbers N such that there exists a
    nonempty proper subset A of N and such that sup A exists in N, but sup A is not in A.
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  2. #2
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    Re: Ordering of a set

    Let B=\{1-1/n\mid n=1,2,\dots\} and let M=B\cup\{1\}. Then B is a nonempty proper subset of M and, with respect to the standard ordering, there exists sup B in M, but sup B is not in B. It is sufficient to define a one-to-one correspondence f : M\to\mathbb{N} and consider the order carried over to \mathbb{N} by f from M.
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  3. #3
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    Re: Ordering of a set

    Thank you. However, I am still a little confused by the order carried over to N by f from M. The set M = {0, 1/2, 2/3, ...} But what kind of function should I use? The idea of (nonstandard) ordering of N confuses me, as I have never encounterd it before.

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  4. #4
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    Re: Ordering of a set

    It's natural to put 1 - 1/n from M into correspondence with n in ℕ, i.e., f(1 - 1/n) = n. The remaining elements are 1 from M and 0 from ℕ, so we pair them up as well, i.e., f(1) = 0. What order ≺ is induced on ℕ? By definition, f(1 - 1/n) ≺ f(1 - 1/m) iff 1 - 1/n < 1 - 1/m, i.e., n ≺ m iff n < m. However, n ≺ 0 for n = 1, 2, ...

    Edit: Corrected n ≺ n to n ≺ m.
    Last edited by emakarov; October 18th 2012 at 09:07 AM.
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