Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: You know you've made it in mathematics when...
Replies: 27   Last Post: Feb 19, 1998 4:16 AM

 Messages: [ Previous | Next ]
 Lee Lady Posts: 24 Registered: 12/12/04
Re: Quick proof of the Well-Ordering Property
Posted: Feb 17, 1998 2:17 PM

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

Date Subject Author
2/14/98 Dave Rusin
2/15/98 Richard Carr
2/15/98 Jeff Erickson
2/15/98 Fickdichhh
2/15/98 Jos Horikx
2/15/98 Dave Rusin
2/17/98 Michael David Cochran
2/17/98 Robin Chapman
2/17/98 Moshe Shulman
2/18/98 Moshe Shulman
2/17/98 Torkel Franzen
2/19/98 Bill Dubuque
2/15/98 Michail Brzitwa
2/16/98 Brian Howie
2/16/98 Matthew P Wiener