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Article 7324 of comp.ai.philosophy:
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>From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Human intelligence vs. Machine intelligence
Message-ID: <1992Oct16.180352.13326@oracorp.com>
Organization: ORA Corporation
Date: Fri, 16 Oct 1992 18:03:52 GMT
Lines: 34

In article <Bvz82n.A7w@cs.bham.ac.uk>, axs@cs.bham.ac.uk (Aaron
Sloman) writes:

>My argument, summarised very briefly in an earlier posting, is that
>even if F is consistent, Penrose is deluded in thinking that he can
>use Godel's theorem as a basis for discovering that the Godel
>sentence G(F) is true. Many of his critics share the same delusion.
>
>The full argument requires very detailed analysis of what the Godel
>formula actually says: I claim it merely says that a very
>complicated arithmetical property belongs to a certain very large
>number. It gets subtly misinterpreted as saying something about a
>certain formula and its relationship to a formal system.

You are right that the Godel formula is a statement of pure
arithmetic, but you are completely wrong to say that it is not also a
statement about formal systems.

Godel's arguments showed that the notion of consistency of a theory
can be "arithmetized": he gives a construction which applied to any
formal system S (expressive enough to interpret Peano arithmetic)
yields a formula G of pure arithmetic such that (1) if S is
consistent, then G is a true statement of arithmetic, and (2) if S is
consistent, then S does not prove G.

Of course, since G is a statement of pure arithmetic, you can ignore
all of the metamathematical implications of G and simply consider it
as a statement about numbers. However, as a statement about numbers,
there is no reason to regard it as very interesting. The whole point
of talking about G is its metamathematical implications.

Daryl McCullough
ORA Corp.
Ithaca, NY


