Newsgroups: comp.ai.philosophy
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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Etheric spiritual forces, are there...?
Message-ID: <1994Sep27.234533.29643@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Tue, 27 Sep 1994 23:45:33 GMT
Lines: 47

lupton@luptonpj.demon.co.uk (Peter Lupton) writes:

>allsop@fc.hp.com (Brent Allsop) writes:
>> 
>> 
>> Eddie asks,
>> 
>> > The really interesting question, though, is WHY is the universe
>> > built in such a way that we cannot have what we want - completeness
>> > and consistency.
>> 
>> 	Godel found a necessary truth.  In other words it applies to
>> all universes and systems not just ours.  You can't make a universe or
>> system that has both completeness and consistency.  Omnipowerful Gods
>> can't change necessary truth!
>
>Goedel's Theorem includes the requirement that the theory in question
>is axiomatic - that is, a theory whose theorems could be enumerated by a
>computer. I am not qualified to give an authoratitive answer, but
>I doubt whether the Gods would be so constrained.

It is true that Goedel was interested in axiomatic theories, and so
proved his theorem for that case. However, the proof method he used is
actually more general, and applies to any theory whatsoever (axiomatic
or not) in which the notion of "theoremhood" is formalizable and the
theory is sufficiently expressive so that syntactic operations on
sentences can be formalized.

So, to the extent that God's beliefs constitute a theory, if God
Himself has a notion of "what God believes", then His beliefs are
either inconsistent or incomplete.

The way that some theories, such as the theory of true arithmetic,
escape from the consequences of Goedel's theorem is by not having a
formalization of theoremhood. There is no theoremhood predicate for
true arithmetic which is expressible in the language of arithmetic.
It is not the fact that arithmetic is non-axiomatic that allows it
to be complete, it is the fact that it is not self-referential---it
cannot "talk" about its own theorems. 

Daryl McCullough
ORA Corp.
Ithaca, NY




