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Re: Quick proof of the Well-Ordering Property
Posted:
Feb 17, 1998 2:17 PM
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In article <34E95F44.5FE6@pitt.edu>,
Michael David Cochran <mdcst22@pitt.edu> wrote:
>Does anyone have a fast proof of the Well-Ordering Property? The
>Well-Ordering Property states that every nonempty set of positive
>integers contains a smallest member.
Since the set S is non-empty, it contains at least one element. Choose
one. For convenience, call it n. Now if n is not itself the smallest
element in S, then the smallest element would be smaller than n. Hence
without loss of generality we may replace S by the subset of S
consisting of integers in S less than or equal to n. But there are
only finitely many positive integers less than or equal to n. (In
fact, there are only n of them!) Therefore, without loss of generality
we may suppose that S is finite. But clearly every finite set of
integers (or real numbers for that matter) has a smallest element.
If you can read pdf (Adobe Acrobat) files, you might be interested in a
rather extended discussion of some related matters in my lengthy article
on the Division Algorithm: www.math.Hawaii.Edu/~lee/courses/Division.pdf.
(Versions of the article in dvi format and postscript format are also
available.)
--
Trying to understand learning by studying schooling
is rather like trying to understand sexuality by studying bordellos.
-- Mary Catherine Bateson, Peripheral Visions
lady@Hawaii.Edu
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