Re: Please explain the "Well Ordering Property"

From: Dave (nospam_at_nowhere.com)
Date: 03/26/05 

Date: Sat, 26 Mar 2005 21:42:29 +0000

Norm Dresner wrote:
> "Dave" <nospam@nowhere.com> wrote in message news:4243eff0@212.67.96.135...
>
>>I have a book on number theory, which is for interest sake only, as I'm
>>not a mathematician, and are not particularly good at maths. In this book
>>(page 6) it gives the following:
>>
>>"Every nonempty set of positive integers has a minimum element. We say
>>the set of positive integers is Well Ordered"
>>
>>I can accept that, as {7,4,4,20} has a minimum of 4.
>>
>>but it then goes on to say
>>
>>"On the other hand, the set of all integers is not well ordered, since
>>there are sets of integers that do not have a smallest element (as the
>>reader should verify"
>>
>>Now the only difference between the set of positive integers (1,2,3 ..)
>>and the set of all integers (...,-3,-2,-1,0,1,2,3) is the fact the
>>latter can contain 0 and negative numbers.
>>
>>
>>Hence I can't see where there is a difference if you define well ordered
>>has having a smallest element.
>>
>>
>
>
> The set of all integers is ordered by the usual (<) ordering. It just isn't
> "Well Ordered" because it doesn't have a least element. That's part of the
> definition of a Well Ordering. While it might not matter that much to you,
> there are some technical mathematical situations in which it makes a
> difference.
>
> Norm
>

Can the list be finite in length?

If the 'non empty list' can have say 3 elements, then surely there is
one minimum element, no matter if each of those three is positive,
negative or a mixture.

Conversely, if the list must contain all the posible integers, then I
can see that a list of all the negative integers, will be infinite in
length, and will not have a minimum. Whereas as list of all positive
integers will have a minimum (which is zero), but it will not have a
maximum element.

I'm still none the wiser.

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