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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Review of Shadows of the Mind
Message-ID: <1995Apr4.022822.15745@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Tue, 4 Apr 1995 02:28:22 GMT
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Xref: glinda.oz.cs.cmu.edu sci.physics:116185 sci.cognitive:7187 comp.ai:28762 sci.philosophy.tech:17387 sci.skeptic:109134

ghrosenb@phil.indiana.edu (Gregg Rosenberg) writes:

I've lost track of the levels of indentation, so I'll simply
annotate with who wrote what:

> 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.

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

>alan>
>alan> I don't know how to take this.
>alan> What about taking all true statements of arithmetic --
>alan> what do you mean by the Godel statement there?


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


>greg> I have a question about your claim here, Daryl. Although it has some
>greg> intuitive appeal, I'm not sure Godel's theorem could apply to 'all
>greg> true statements of arithmetic.' Two problems: The Godel sentence
>greg> for a theory is itself a true statement of arithmetic, and there
>greg> are too many true statements of arithmetic to form a set. Their
>greg> size means that they could not be the domain of any formal theory
>greg> as 'formal theory' is classically used. I am not sure how you
>greg> are envisioning applying Godel's result to the ???class??? of statements
>greg> in question. Am I missing something?

Maybe. The set of true statements of arithmetic is a countable set, it
just isn't recursively enumerable, meaning that there is no algorithm
which will generate the true statements and only the true statements.
It isn't because the set of true statements is "too big", however--it
is because it is too complicated. The set of true statements is a
subset of the set of *all* statements (true and false), and the latter set
is recursively enumerable. (As a matter of fact, Godel's coding system
enumerates it.)


>daryl> are recursively enumerable theories. That isn't true. For example,
>daryl> take the union of true arithmetic (interpreted as a collection of
>daryl> statements about the finite ordinals) with ZFC. The resulting theory
>daryl> is non-r.e., but it has a Godel statement, and is incomplete.

>greg> Hmmmm, my gut reaction is that we cannot take the union of ZFC and true
>greg> arithmetic, because the result will just be too big....

No, it's not too big.

Daryl McCullough
ORA Corp.
Ithaca, NY

