
Re: Quick proof of the WellOrdering 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 WellOrdering Property? The >WellOrdering Property states that every nonempty set of positive >integers contains a smallest member.
Since the set S is nonempty, 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

