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From: gwyn@brl-vgr.ARPA (Doug Gwyn )
Newsgroups: net.math
Subject: Re: INTERESTING NUMBERS
Message-ID: <1411@brl-vgr.ARPA>
Date: Sat, 12-May-84 00:50:29 EDT
Article-I.D.: brl-vgr.1411
Posted: Sat May 12 00:50:29 1984
Date-Received: Wed, 9-May-84 03:25:54 EDT
References: <273@nbs-amrf.UUCP>
Organization: Ballistics Research Lab
Lines: 39

I was sort of hoping that someone would try to seriously analyze the
flaw in the "proof" that there are no uninteresting numbers.  For
what it's worth, here's my two cents' worth:

The "proof" in essence is:
	(1) For purposes of proof-by-contradiction, assume that there
	are more than 0 uninteresting numbers.
	(2) Let U = { n : n is an uninteresting number }.
	U is non-empty by assumption (1).
	(3) Since the n are positive integers (not stated before,
	but since it is important to the "proof" we will now make
	this restriction), the set U must have a least member n0.
	(4) Since n0 = n : n is the minimum of the set U, n0 is
	interesting.
	(5) But all n are partitioned as either interesting or
	uninteresting, so no n can be both.
	(6) This contradiction in the classification of n0 implies
	that the hypothesis (1) is false, i.e. there are no
	uninteresting numbers.  Q.E.D.

I find that two points come immediately to mind when analyzing this
"proof":
	(a) The definition of "interesting" is fuzzy.  Apparently it
	is sufficiently general to include "is the least member of a
	well-defined set" or we would not be able to conclude (4).
	(b) In which case, we have another Russell's antinomy here,
	in which the definition of U is such as to make U impossible
	to construct.

I conclude that this is just another instance of the general problem
of proper definition of classes of objects; there are several points
of view on the proper resolution of these.  My claim would be that one
cannot properly define the predicate "is an interesting number" unless
it excludes "is the least member of a well-defined set" (by which I
mean a level-0 class, for those who subscribe to the 2-levels-of-classes
approach).

More discussion, please, particularly from mathematicians who have
some other way of resolving the antinomy.