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From: jqb@netcom.com (Jim Balter)
Subject: Re: Putnam reviews Penrose.
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References: <3t0tn4$p32@netnews.upenn.edu> <3tqldi$bvb@saba.info.ucla.edu> <3tqm3n$l5p@agate.berkeley.edu> <95Jul10.044337edt.6061@neat.cs.toronto.edu>
Date: Mon, 10 Jul 1995 13:46:34 GMT
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In article <95Jul10.044337edt.6061@neat.cs.toronto.edu>,
Calvin Bruce Ostrum <cbo@cs.toronto.edu> wrote:
>In article <3tqm3n$l5p@agate.berkeley.edu>,
>   Edward Faith  <epfaith@aol.com> wrote:
>
>| zeleny@oak.math.ucla.edu (Michael Zeleny) wrote:
>| >cbo@cs.toronto.edu (Calvin Bruce Ostrum) writes:
>| 
>| >>Goedel's theorem is a *theorem*.  No reasoning system
>| >>whatsoever can escape it, and it's that simple.  No reasoning system
>| >>whatsoever can exhibit a proof of a Goedel sentence for a formal system
>| >>F in that system.  There *is* no such proof, after all.  
>| >
>| >Only if your conception of proof is sufficiently impoverished to
>| >exclude the sort of informal reasoning that enables us to see the
>| >Goedel sentence as expressing a truth.
>| 
>| I think Mr. Ostrum meant a proof *in the formal system F*.
>| Not any old proof.  Here's my rephrasing of the sentence:
>| 
>| No reasoning system whatsoever can exhibit a proof in a
>| formal system F of a Goedel sentence for that system.
>
>Yes, I should think that would be completely obvious to anyone
>who wasn't merely trying to cause trouble.

Are you implying that Mr. Zeleny, with his vast store of knowledge,
would stoop to such levels?

>Note that when I
>say there is "no *such* proof" (emphasis added), the "such" 
>refers back to an earlier characterisation of the type of proof.
>And what type is that?  The type referred to in the immediately
>preceding sentence, i.e., "a proof in *that* system" (emphasis
>added).  And what does "that system" refer to?  Clearly, it refers
>to system F.  Isn't this perfectly clear?  I mean, am I using language
>incorrectly here?  Maybe I don't know what "such" means, or how
>to use it?  Can't I say "He was looking for a unicorn, but there
>is no such thing", for example?   Am I forced always to say "He was
>looking for a unicorn, but there are no unicorns"?   I don't get it.
>
>To repeat the original point, the problem with these attempts to
>show man surpasses machine is that they equivocate on the nature of
>proof.  They insist that a machine use the "impoverished" formal
>notion of proof, whereas man is entitled to use a richer "informal"
>kind of proof. 

Um, yes, some might call it hypocrisy.

-- 
<J Q B>

