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From: jqb@netcom.com (Jim Balter)
Subject: Re: rereRe: The end of god
Message-ID: <jqbCyo27J.1vr@netcom.com>
Organization: NETCOM On-line Communication Services (408 261-4700 guest)
References: <Cxzo7E.91v@gpu.utcc.utoronto.ca> <Harmon.776.000A2404@psyvax.psy.utexas.edu> <Cy72p4.B1r@gpu.utcc.utoronto.ca> <1994Oct25.052916.3600@gov.nt.ca>
Date: Thu, 3 Nov 1994 00:47:42 GMT
Lines: 28

In article <1994Oct25.052916.3600@gov.nt.ca>,
Graham Monroe <gmonro@gov.nt.ca> wrote:
>I may be diverging from your conversation somewhat (I haven't been
>consistently following the thread) but as I understand the Goedel
>theorem, there are, in any sufficiently powerful logical system,
>statements that can neither be proven true nor false. This means they
>have no truth value and the statements, or their negations, can be
>taken as axioms. Asking whether the statement is true or false is
>in effect asking a meaningless question.

Suppose there is a consistent axiomatic system S with sufficient
expressibility to encode the proposition "S is consistent".  Then, by Goedel,
that proposition cannot be proved within S.  Yet it most certainly is true and
is not false.  It can be proved in some system greater than S.  If it could
not be, then "Suppose there is a consistent axiomatic system S ..." would be
no more meaningful than "Suppose there is a blue axiomatic system S ...".

AI programs running on computers, being based upon axiomatic systems, either
cannot prove that their operational axioms are consistent or, if they can, are
mistaken.  Of course humans, whether based upon axiomatic systems or not, have
a similar limitation.  (Actually, I think we can count on any human to be
inconsistent.  A trivial sort of inconsistency can be seen by asking a proof
reader to proof a 1000 page manuscript.  After proofing each page, ask whether
there are any uncorrected errors on that page, and the heartfelt answer will
be "no".  After finishing the entire work, ask whether there remain any
uncorrected errors, and the response will be "of course".)
-- 
<J Q B>
