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Constructing Well Ordered Set

  1. Feb 17, 2015 #1
    1. The problem statement, all variables and given/known data
    My teacher has notes online that say:

    A Simple Construction Technique for WellFounded Orders
    Any function ƒ : S→N defines a wellfounded order on S by
    x < y iff ƒ(x) < ƒ(y).

    Example:
    Lists are wellfounded by length. Binary trees are wellfounded by depth, by number of nodes, or by number of leaves. ℤ is wellfounded by absolute value.
    Derivations for a grammar are wellfounded by length. These orders are nonlinear.

    I am having trouble understanding how these are non-linear orders. Particularly "ℤ is wellfounded by absolute value". From my understanding a linear order is where each element in the set is comparable to the other elements of the set.

    So for the abs of ℤ the order I think we get is:
    0, -1, 1, -2, 2, -3, 3, ...

    which seems like -1 < -2 < 3 and so on because of the way we defined it and thus they are comparable. Am I misunderstanding something in this?
     
  2. jcsd
  3. Feb 18, 2015 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Right.
    The usual ordering of Z is not relevant here.
     
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