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From: gmf@uvacs.UUCP
Newsgroups: net.math
Subject: Interesting Numbers
Message-ID: <1279@uvacs.UUCP>
Date: Sat, 5-May-84 10:25:42 EDT
Article-I.D.: uvacs.1279
Posted: Sat May  5 10:25:42 1984
Date-Received: Sun, 6-May-84 07:38:18 EDT
Lines: 25


>> But the "proof" that there is no "uninteresting" number by
>> emiminating the smallest such one at a time is invalid

I'm not sure my "proof" is being referred to, but maybe I ought
to expand it:

Let U be the set of uninteresting numbers, and suppose (by way of
contradiction) that U is non-empty.  Then U has a least element x
(by a principle equivalent to the axiom of mathematical induction).
Then x is uninteresting by virtue of being in U.  But surely x is
interesting by virtue of being the least uninteresting positive
integer (like 1729 is interesting by virtue of being the least
positive integer which is the sum of 2 cubes in 2 different ways).
Thus x is both uninteresting and interesting, a contradiction.  This
arose from assuming U non-empty.  Thus U, the set of uninteresting
positive integers, is empty.  Thus all positive integers are interesting.


To call this proof "invalid" is to call the axiom of mathematical
induction invalid.  A proof by mathematical induction does not proceed
by considering one case at a time until all cases are taken care of.
This is only possible for finite sets.

     Gordon Fisher