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Article 5386 of comp.ai.philosophy:
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>From: maione@cs.toronto.edu (Ian Christopher Maione)
Subject: Comments on Godel's Theorem
Message-ID: <92May3.223926edt.47925@neat.cs.toronto.edu>
Organization: Department of Computer Science, University of Toronto
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Date: 4 May 92 02:39:57 GMT
Lines: 70

     A couple of comments on Godel's theorem:

   This is just my personal opinion, but I think in the case of Godel's
results, or any of the more advanced theorems in first-order logic
(i.e Skolem-Lowenheim's theorems, etc) it's really best to learn them
in full gory detail.  Of course, having a book which also explains things
in a more informal way at the same time is good, but if you have only
the latter, it's quite easy to convince yourself you understand the
theorem, when in fact there are large gaps in your understanding.  I
experienced this myself when I was exposed to an informal approach in
a philosophy class, and a rigorous approach in a mathematics course.
The former was nearly devoid of content, while I understood the latter
fine.  Some of the crucial aspects of the theorem, especially when
things are happening simulatneously on the semantic and syntactic levels
require precision.  Another crucial aspect of the theorem is the fact
that the proof of the first incompleteness theorem can be formalized
within the formal system you are working in.  Although often glossed over
in informal accounts, this fact is what really gives the theorem its
significance, and leads you to the second incompleteness theorem.
Besides, the proof itself is a beautiful thing, which should be
appreciated in its fullest if possible.

    In another posting, it was suggested that the self-reference in
Godel's original sentence represents something which is "inherently"
true regardless of the interpretation.  The "meaning" of Godel's
sentence is dependent on the Godel-numbering you have set up,  and it
is possible to come up with an interpretation which makes it false.  By
construction, though, it IS true in the standard model of arithmetic.
Another way of proving the theorem is to (essentially) formalize the
halting problem within arithmetic.  This undecidability of the halting
problem ends up giving you your undecidable sentence.

   A member of the mathematics department here at Toronto told me the
following interesting anecdote:  apparently there is a mathematician in
Holland (sorry, forget his name),  who every couple of years publishes
a detailed proof that arithmetic is consistent.  This typically leads
to a frantic search for a couple of months for an error in the proof,
which to date has always been found.  Now this fellow is no fool - he
knows what Godel's second theorem says.  The point is though,
that if there were such a proof, then that would be a proof that
arithmetic is in fact INconsistent!

    There is a paper by J.R Lucas, called "Minds, Machines and Godel",
which pretty much puts forward a standard account of why Godel's
theorem (allegedly) refutes strong AI.  In fact the argument is quite
flawed, and if anyone wants to understand the relation of Godel's theorem
to AI, it would be good to start with it.  If you can pick out the flaws
in Lucas' argument, you've probably got a good grasp of the issues.  I
personally feel that Godel's theorem is pretty much irrelevant to AI,
both from a practical and philosophical standpoint.  The problem with
about every argument based on Godel's theorem against AI is that
somewhere an unwarranted assumption is always snuck in about the
capabilities of human beings, which on close examination doesn't hold up.
This is (one) of the mistakes Lucas makes, and as Marvin Minsky pointed
out, Penrose makes the same mistake.
    The reference for Lucas' paper is
   "Minds, Machines and Godel"  Philosophy Vol XXXVI (1961) 112-227

   By the way, someone also mentioned a book by Hao Wang called
"Conversations with Kurt Godel".  I read "Reflections on Kurt Godel",
and a second book was mentioned, but I've never been able to find it.
If it does in fact exist, and someone can point me to it, please email
me.

Regards,
Ian
"Birds are living dinosaurs, we can substitute the term 'dinosaur' for
 'bird'; A dinosaur in the hand is worth two in the bush"
Gautier, J.A & Padian, K "The Origin of Birds and the Evolution of Flight"



