Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10 5/3/83; site westcsr.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!mhuxl!houxm!hogpc!houti!ariel!vax135!ukc!west44!westcsr!pkelly
From: pkelly@westcsr.UUCP
Newsgroups: net.math
Subject: Re: Interesting number proof invalid
Message-ID: <130@westcsr.UUCP>
Date: Thu, 17-May-84 10:45:47 EDT
Article-I.D.: westcsr.130
Posted: Thu May 17 10:45:47 1984
Date-Received: Sat, 12-May-84 09:42:16 EDT
References: <7672@watmath.UUCP> <2457@allegra.UUCP>
Organization: CS Dept., Westfield College, London
Lines: 25

We have a proof that all numbers, for example 1,2,3... are interesting
since any set of non-interesting number drawn form the number set must
have a least element, which will be interesting if only for that reason.

If, for a moment, we accept this, and disregard the reasonable objections
that (1) this is silly, and (2) the induction implied doesn't work since 
once you choose your next 'interesting' set to exclude the least element of the 
previous one, you make that element boring again.

We can then note that the argument doesn't follow for reals, leaving open the
assertion that real numbers are proportionately less exciting than rational,
naturals etc (constructivists who might up to now have shuddered at the naivete
of the discussion can now cheer).
   The reason is that a set of real numbers can be bounded below without
containing its bound - there might be an infinite progression of decreasing 
boring numbers (perish the thought), which neither stop decreasing nor 
reach any chosen value without eventually crossing it.
   The argument follows, of course, for lots of other uncountable sets.

		Yours, Paul Kelly.

PS 
 The rationals example takes a little explanation - instead of taking the 
least el. as your new interesting value, take the first in some enumeration
(diagonalisation) of the rationals.