From newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.ecf!utgpu!watserv1!watdragon!logos.waterloo.edu!wlfong Wed Apr 22 12:04:23 EDT 1992
Article 5178 of comp.ai.philosophy:
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>From: wlfong@logos.waterloo.edu (Philip W. L. Fong)
Subject: Godel Incompleteness Theorm
Message-ID: <1992Apr22.010037.21305@watdragon.waterloo.edu>
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Date: Wed, 22 Apr 1992 01:00:37 GMT
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Recently I posted a request for introductory references on Godel Incompleteness
Theorm.  I received a lot of replies, and I would like to thank
all who reponded to the posting.  Attached is a list of the replies I
received.  Some of them contain bibliographical information that you
may be interested in.  If you would like to suggest more introductory
references on Godel's Theorm, please forward them to me at
wlfong@logos.waterloo.edu.  Thank you again.

Philip

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Philip Fong                                         wlfong@logos.waterloo.edu
University of Waterloo                                          (519)888-4674
Ontario, Canada                                                 (519)725-7795
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Date: Wed, 15 Apr 1992 16:07:38 -0400
>From: Gary Forbis <forbis@u.washington.edu>

In article <1992Apr15.160747.9350@watdragon.waterloo.edu> you write:
>I am interested in knowing more about Godel's Incompleteness Theorm, especially
>the proof and its implication on the limitation of human/artificial 
>intelligence.  However I am not a theory person and I don't think I will
>understand Godel's original document.  Is there any easy-to-comprehend 
>text/reference (for an average CS student) which covers the proof and 
>discusses the implication of the theory?
>
>Philip

I don't have the proof completely in my head but can give you several related
proofs.

Euclid's proof that there is no greatest prime goes:

1.  Suppose that there is some greatest prime.
2.  Multiply all of the primes less than or equal to that prime together
    and add 1.
3.  Becuase this new number is not divisible by any of the primes and is
    greater than the supposed greatest prime either it is prime or the multiple
    of primes greater than the supposed greatest prime.
4.  1. is false.

I don't remember the name of the theorem that states there are orders of
infinities but here is the proof that this is so.

1.  Suppose there is some ordered set N which contains all of the real numbers
    x such that 0<x<=1.
2.  Construct a number as follows:
    a.  look at the nth digit of the nth element in the ordered set.
    b.  if that digit is a zero put a nine in the nth digit of the new number.
    c.  if that digit is not a zero put a zero in the nth digit of the new
        number.
3.  For all n the constructed number differs from the nth number in the 
    ordered set in at least the nth digit after the decimal point.
4.  1. is false.

I'm having problems remembering Goedel's incompleteness theorem but it follows
the above examples.  It relies upon a map of the axioms of integer arithmetic
to prime numbers and theorems to non-prime numbers.  Because there are an
infinite number of primes there are an infinite number of axioms to integer
arithmetic.

It turns out that any formal system falls prey to the same technique.
This is why many claim intelligence is not formalizable and humans are
not computable.

It is an open question wheather or not humans have finite or countably
infinite capabilities or uncountably infinite capabilities.

Any intermediate logic book should have a version of Goedel's incompleteness
theorem.

I'm sorry I can't be of more help.

--gary forbis@u.washington.edu

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Date: Wed, 15 Apr 1992 16:28:36 -0400
>From: kim@unix1.cs.umass.edu (John Kim)

In article <1992Apr15.160747.9350@watdragon.waterloo.edu> you write:
>I am interested in knowing more about Godel's Incompleteness Theorm, especially
>the proof and its implication on the limitation of human/artificial 
>intelligence.  However I am not a theory person and I don't think I will
>understand Godel's original document.  Is there any easy-to-comprehend 
>text/reference (for an average CS student) which covers the proof and 
>discusses the implication of the theory?
>
>Philip

_Godel, Escher, Bach_


-- 
                            
   John Kim               Portate mal porque la vida es demasiado 
                          corta para portarse bien. 
                                                       (INDI)

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Date: Wed, 15 Apr 1992 16:42:28 -0400
>From: Christopher Scott Weyand <weyand@skinner.cs.uoregon.edu>

   I am interested in knowing more about Godel's Incompleteness Theorm, especially
   the proof and its implication on the limitation of human/artificial 
   intelligence.  However I am not a theory person and I don't think I will
   understand Godel's original document.  Is there any easy-to-comprehend 
   text/reference (for an average CS student) which covers the proof and 
   discusses the implication of the theory?

   Philip

See Godel, Escher, Bach (An Eternal Golden Braid) by Douglas Hofstadter.
An excellent book.

Chris Weyand
weyand@cs.uoregon.edu

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>From: sharder@cogsci.edinburgh.ac.uk
Date: Thu, 16 Apr 1992 09:00:53 -0400

Introductions to Godels theorem can be found in
Nagel and Newman(1959): Godels proof (New York University Press)
Nagel and Newman(1956): 'Godels proof' in Scientific American,
reprinted in Copi and Gould (1967): Contemporary readings in Logical Theory.

The Penrose/Lucas argument can be found in
Penrose: The Emperors New Mind
Lucas(1961): 'Minds, Machines and Godel' in Philosophy 36,p.112-127
Lucas(1970): The Freedom of Will (Clarendon Press, Oxford)

The Lucas-argument is in my taste much better than Penrose's. I
haven't read the Lucas article, but the book is good. You can skip the
first couple of hundred pages about philosophical pros and contras of
determinism (I did).

Counterarguments to Lucas:
There are a lot; I particularily recommend
Hadley(1987): 'Godel, Lucas and mechanical models of the mind' in
Computational intelligence vol.3 no.2
as it is the most recent, and gives a good overview.
Another central text is 
Putnam(1961): 'Minds and Machines'. In Hook(ed.):Dimensions of Mind
and in Anderson(ed.)(1964): 'Minds and Machines'. 
His argument only takes up one page or so (in the beginning of the article).

Hope this helps,

Soren Harder

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Soren Harder, (MSc student)
Centre for Cognitive Science, 2 Buccleuch Place, Edinburgh
E-mail: sharder@cogsci.ed.ac.uk
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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>From: bill@NSMA.AriZonA.EdU (Bill Skaggs)
Date: 15 Apr 92 23:57:36 GMT

In article <1992Apr15.160747.9350@watdragon.waterloo.edu> 
wlfong@logos.waterloo.edu (Philip W. L. Fong) writes:
>      Is there any easy-to-comprehend 
>text/reference (for an average CS student) which covers [Godel's] proof 
>and discusses the implication of the theory?

  Far and away the best thing to read is Hofstadter's "Goedel,
Escher, Bach: An Eternal Golden Braid."

	-- Bill

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>From: atten@phil.ruu.nl (Mark van Atten)
Date: 16 Apr 92 14:17:03 GMT

wlfong@logos.waterloo.edu (Philip W. L. Fong) writes:

>I am interested in knowing more about Godel's Incompleteness Theorm, especially
>the proof and its implication on the limitation of human/artificial 
>intelligence.  However I am not a theory person and I don't think I will
>understand Godel's original document.  Is there any easy-to-comprehend 
>text/reference (for an average CS student) which covers the proof and 
>discusses the implication of the theory?

>Philip

The best explanation I have found is in 'Infinitiy and the Mind ', by Rudy
Rucker. It contains both an explanation by analogy (story) and a technical
(hence more precise) explanation (also easy to comprehend, but take your time)

>From Goedel's original paper, I recommend the non-technical introduction.

Contrary to general opinion, I do not believe Hofstadter's book is the best
introduction or explanation of Goedel's argument. But once you grasp the
argument, read it for some interesting suggestions. But it is too general.

A very good discussion of the use of mathematical/logical results in arguments
about human/artificial intelligence can be found in 'From mathematics to
philosophy' by Hao Wang.

Best wishes,

Mark.

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Date: Thu, 16 Apr 1992 13:39:00 -0400
>From: Nollaig MacKenzie <GL250011@Orion.YorkU.CA>

Greetings,

You've probably already got this reference, but...

Nagel and Newman's _Godel's Proof_ is still a pretty good 
intro to the thing.

There are three things liberal arts majors should never hear 
about unless they take a whole year course in them:

	Godel's Incompleteness Theorem

	The Uncertainty Principle

	The Special Theory of Relativity

Cheers, N.

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Date: Fri, 17 Apr 1992 12:11:02 -0400
>From: tkprasad@valhalla.cs.wright.edu (Thirunarayan Krishnaprasad)


You can read "Godel Escher Bach" by Douglas Hofstader
and "Godel's Proof" by Nagel and ...
Both have extremely informal and palatable treatment of
Incompleteness theorem. The former is a Pulitzer Prize Winner.
The latter book was written in 1950's and is NYU Press I think.
You can also look at "Metamagical Themas" by the first author
for an article on self-reference. 

Avoid looking at First-order logic text books. Even though I work in
logic, I understood more from the informal descriptions than the gory
proof in Enderton!

Good Luck.
Prasad

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>From: <ld231782@longs.lance.colostate.edu>
Date: Sat, 18 Apr 1992 01:51:12 -0400

--
One book is "Godel's Theorem" by Nagel & Newman. A bit dated, but it is
very thorough for a beginner.  Not too philosophical but it can give you
an idea why some think the theorem applies to machine thought. Roughly the
argument is that machines are limited by the same theorem (Turing proved
that) so that there are some things that simply can't be computed. Yet there
seems to be no limitation on our thoughts.


ld231782@longs.LANCE.ColoState.EDU

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>From: peter@sysnext.library.upenn.edu (Peter C. Gorman)
Date: 17 Apr 92 16:48:12 GMT

In article <1992Apr15.160747.9350@watdragon.waterloo.edu>  
wlfong@logos.waterloo.edu (Philip W. L. Fong) writes:

> ... Is there any easy-to-comprehend 
> text/reference (for an average CS student) which covers the proof and 
> discusses the implication of the theory?

I highly recommend the following:

Nagel, Ernest, and James R. Newman.  Godel's Proof.  New York: NYU Press,  
1958.  118 pp.

This is very readable, but doesn't skimp on the details relevant to the  
non-mathematician.  It gives a very good account of the historical and  
philosophical environment of the Proof, and is written in a clear and  
direct style.  I happened to find it on a remainders table in a bookstore  
for $1.  One of the best dollars I ever spent.

--
Peter Gorman
University of Pennsylvania
Library Systems Office
peter@sysnext.library.upenn.edu

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Date: Mon, 20 Apr 1992 08:43:54 -0400
>From: Gordon Joly <G.Joly@cs.ucl.ac.uk>

A good introduction may be found in "Forever Undecided -
A Puzzle Guide to Godel" Raymond Smullyan, Oxford University
Press, 1987.


Gordon Joly                                       +44 71 387 7050 ext 3703
Internet: G.Joly@cs.ucl.ac.uk        UUCP: ...!{uunet,uknet}!ucl-cs!G.Joly
Computer Science, University College London, Gower Street, LONDON WC1E 6BT

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>From: oz@ursa.sis.yorku.ca (Ozan Yigit)
Date: Tue, 21 Apr 1992 02:32:53 GMT

As an addition to the previously mentioned refs on Goedel, there
is also a book by Stuart Shanker (ed) "Goedel's Theorem in Focus"
[Croom Helm, New York, 1988] that may be of some interest. These
are the articles included in the book:

I.	John W. Dawson, Jr.: Kurt Goedel in Sharper Focus
II.	Kurt Goedel: On Formally Undecidable Propositions of Principia
	Mathematica and Related Systems I (1931)
III.	Stephen C. Kleene: The Work of Kurt Goedel
IV.	John W. Dawson, Jr.: The Reception of Goedel's Incompleteness
	Theorems
V.	Solomon Feferman: Kurt Goedel: Conviction and Caution
VI.	Michael D. Resnik: On the Philosophical Significance of
	Consistency Proofs
VII.	Michael Detlefsen: On Interpreting Goedel's Second Theorem
VIII.	S. G. Shanker: Wittgenstein's Remarks on the Significance
	of Goedel's Theorem

enjoy.	oz
---
Information is in the mind of  |  internet: oz@nexus.yorku.ca
the beholder. - R. Jackendoff  |  phone:[416] 736 2100 x33976

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>From: e343mh@tamuts.tamu.edu (Michael Hand)
Date: 21 Apr 92 16:46:19 GMT

oz@ursa.sis.yorku.ca (Ozan Yigit) writes:

>As an addition to the previously mentioned refs on Goedel, there is also a 
>book by Stuart Shanker (ed) "Goedel's Theorem in Focus" that may be of some 
>interest. These are the articles included in the book:
>...
>VIII.	Shanker: Wittgenstein's Remarks on the Significance of Goedel's Theorem

Shanker's paper is the only paper that is not reprinted there from a 
previous publication, I think.  It's huge - about 100 pages, as I recall.
The volume seems peculiar: in the presence of rather famous papers on 
Goedel's results, Shanker's paper makes its first appearance.  Think this 
is because the author of the paper is a close friend of the volume's editor?
  ;->
Michael

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>From: G.Joly@cs.ucl.ac.uk (Gordon Joly)
Date: 21 Apr 92 15:22:11 GMT

> In article <1992Apr15.160747.9350@watdragon.waterloo.edu> 
> wlfong@logos.waterloo.edu (Philip W. L. Fong) writes:
> >      Is there any easy-to-comprehend 
> >text/reference (for an average CS student) which covers [Godel's] proof 
> >and discusses the implication of the theory?
> 
>   Far and away the best thing to read is Hofstadter's "Goedel,
> Escher, Bach: An Eternal Golden Braid."
> 
> 	-- Bill

Well, first have a skim through this, then read one of the other texts
that have been mentioned, then go back to GEB. Then, you will be as
ready to meet your maker:-) Also, you may need this word as part of
your study:

DEFINE grok
DEFINITION 0
grok \'gro-\ \'gru:\ \'gro-w\ \'gro-(-*)r\ vb or grokked;  or grok.ing
   [ME groken, fr. OE gro-kan; akin to OHG gruokan] 1: to think in a
   certain way - possibly `GROK - Grok's Recurisve OK?'' (obscure) 2:
   a new departure in thought.

Gordon.
____

Gordon Joly                                       +44 71 387 7050 ext 3703
Internet: G.Joly@cs.ucl.ac.uk        UUCP: ...!{uunet,uknet}!ucl-cs!G.Joly
Computer Science, University College London, Gower Street, LONDON WC1E 6BT

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