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From: torkel@sics.se (Torkel Franzen)
Newsgroups: comp.ai.philosophy
Subject: Re: Minds, machines, and Godel
Message-ID: <1991Jan16.182120.20961@sics.se>
Date: 16 Jan 91 18:21:20 GMT
References: <1991Jan16.035058.7465@bronze.ucs.indiana.edu>
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Organization: Swedish Institute of Computer Science, Kista
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In-Reply-To: chalmers@bronze.ucs.indiana.edu's message of 16 Jan 91 03:50:58 GMT

In article <1991Jan16.035058.7465@bronze.ucs.indiana.edu> chalmers@bronze.ucs.
indiana.edu (David Chalmers) writes:

 >Just to refresh your memory, the argument goes like this: if I were a
 >particular Turing Machine T, there would be a mathematical sentence G (the
 >"Godel sentence" of T) that I could not prove.  But in fact I can see that G
 >must be true.  Therefore I cannot be T.  This holds for all T, therefore I am
 >not a Turing machine.

  There is indeed a specific oversight in this argument, contained in the
statement "But in fact I can see that G must be true." For simplicity, let's
stick to arithmetical sentences, where an interpretation is agreed on.
In general we have no idea, given an arithmetical theory T, whether or not
its Godel sentence is true. What we know is that the Godel sentence is true
if the theory is consistent. This is as a rule provable in the theory itself.

  So, to make this a bit more concrete: suppose I present you with a
formal system (or, if you like, a Turing machine generating sentences)
and claim that this system is consistent, and every arithmetical
sentence you can prove is provable in this system. (So as far as
arithmetic is concerned, you're a machine subordinate to or perhaps
identical with this machine.) You can't refute this using Godel's
theorem unless you can prove the consistency of the system. This we
have no reason to believe that you can do. (Consider e.g. the
arithmetical part of ZFC + 'there is an uncountable measurable
cardinal').

  On the other hand it is correct that if we recognize that a formal system
T only embodies valid principles, Godel's theorem provides a means of
extending T to a stronger theory only embodying valid principles. But this
is a different matter.