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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Review of Shadows of the Mind
Message-ID: <1995Mar31.232753.18376@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Fri, 31 Mar 1995 23:27:53 GMT
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Xref: glinda.oz.cs.cmu.edu sci.physics:115840 sci.cognitive:7133 comp.ai:28683 sci.philosophy.tech:17348 sci.skeptic:108743

Alan Smaill <smaill@dcs.ed.ac.uk> writes:

>daryl@oracorp.com (Daryl McCullough) writes:
>
>daryl> Penrose' mistake is in thinking that Godel's incompleteness
>daryl> theorem only applies to computable theories. It applies to a
>daryl> much larger collection of theories than that. Basically, no
>daryl> consistent theory (whether or computable or not) can have as a
>daryl> consequence that it is itself consistent.
>
>I don't know how to take this.
>What about taking all true statements of arithmetic --
>what do you mean by the Godel statement there?

What I said applies to true arithmetic as well as recursively
enumerable theories. The consistency of true arithmetic is *not* a
consequence of true arithmetic, for the simple reason that the
statement "true arithmetic is consistent" is not expressible in
arithmetic.

My claim is that no consistent system (at least none as powerful as
arithmetic) is capable of proving its own consistency. There can be
two different reasons for this: (1) It cannot even *express* the claim
that it is itself consistent, or (2), it can state the claim, but
cannot prove it. Many people think that the only theories in category
(2) (those that can express their consistency, but cannot prove it)
are recursively enumerable theories. That isn't true. For example,
take the union of true arithmetic (interpreted as a collection of
statements about the finite ordinals) with ZFC. The resulting theory
is non-r.e., but it has a Godel statement, and is incomplete.

Daryl McCullough
ORA Corp.
Ithaca, NY
