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From: jim@ism780.UUCP
Newsgroups: net.math
Subject: Re: Orphaned Response - (nf)
Message-ID: <172@ism780.UUCP>
Date: Wed, 16-May-84 00:38:14 EDT
Article-I.D.: ism780.172
Posted: Wed May 16 00:38:14 1984
Date-Received: Thu, 17-May-84 03:51:05 EDT
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#R:uvacs:-127900:ism780:20100003:177600:2359
ism780!jim    May 14 19:04:00 1984

>        (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.

Nonsense.  The fact that a set, when constructed, is empty does not make
the set impossible to construct!  The problem with Russell's "the set of
all sets which do not contain themselves" is that it is semantically
meaningless because the mere introduction of such a "thing" into the language
gives rise to a contradiction (that it neither does nor does not contain
itself).  The notion of an [un]interesting number is much different.
One reasonable definition of an interesting number is "any number which
any person claims to find interesting".  This is perfectly consistent
and meaningful, although it makes a decision procedure rather difficult.

> 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"

Properly?  You are going to have a harder time defining "properly" than
"interesting".  A basic rule for philosophers is that, when formulating
definitions, if their definition excludes cases commonly held to be
members of the set, or includes members commonly held not to be members
of the set, they are making an error.

The real problem is that interestingness is not an absolute quality;
even if we grant that the smallest number which is not interesting
for some other reason *is* interesting just for that reason, it isn't
*very* interesting, and the next number which is not interesting for
any other reason is even less interesting, and in fact we might well
want to concede that it isn't interesting at all.  So, in fact,
while the property of being the least member of a well-defined set
makes a number interesting for certain sets, it obviously does not
do so for all sets.  But the "proof" asserts that the least member of the
set of uninteresting numbers really is interesting, which can be considered
a false claim when several other numbers have already been labeled
as not uninteresting for just the same reason.

-- Jim Balter, Interactive Systems (ima!jim)