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Math Help - Well ordering principle

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    Well ordering principle

    This is the well ordering principle as it is given in Thomas Koshy's number theory book -

    Every non empty set of positive integers has a least element.

    For example the set {17,23,5,18,13} has a least element 5.

    By virtue of the well ordering principle, the set if positive integers in well-ordered. You may notice that the set of negative integers is not well ordered.

    I didn't understand the statement - "You may notice that the set of negative integers is not well ordered".

    A set of negative numbers also has a least number. Or is it that the well ordered principle is confined to only positive numbers and that's the reason the above statement holds good?
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  2. #2
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    Quote Originally Posted by scw_0611 View Post
    Every non empty set of positive integers has a least element.
    I didn't understand the statement - "You may notice that the set of negative integers is not well ordered".
    Consider the set of negative even integers; does it have a least number in it?
    Certainly the set of positive even integers has a least element, 2.
    So while the positive integers are well ordered, the negative integers are not.
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  3. #3
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    Quote Originally Posted by Plato View Post
    So while the positive integers are well ordered, the negative integers are not.
    In the traditional ordering system we use for the integers. It is possible to set up an ordering of the integers where the entire set does have a least element. (But then we abandon the concept of positive and negative. I have never heard of an actual use for such an ordering.)

    -Dan
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    Quote Originally Posted by topsquark View Post
    It is possible to set up an ordering of the integers where the entire set does have a least element. (But then we abandon the concept of positive and negative. I have never heard of an actual use for such an ordering.)
    Extreme care must be taken in putting this.
    But if we accept the Axiom of Choice then one of its consequences/equivalences is the theorem that any set can be well ordered. But be careful, that theorem simply states: Given any set A there exists an order relation G such that G well orders A. Note that we have no way of knowing what G looks like in general. It is likely not \le.
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