Newsgroups: sci.logic,comp.ai.philosophy
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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Penrose' new book
Message-ID: <1994Oct26.174113.5176@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Wed, 26 Oct 1994 17:41:13 GMT
Lines: 28
Xref: glinda.oz.cs.cmu.edu sci.logic:8722 comp.ai.philosophy:21395

"Raymond J Nawara Jr." <dreck@rahul.net> writes:

>daryl@oracorp.com (Daryl McCullough) writes:
>
>> 4. The collection of "unassailable beliefs" of Penrose. Assuming
>> that this collection is closed under logical deduction, it cannot
>> be consistent and include the statement "The collection of
>> Penrose' unassailable beliefs is consistent."
>
>No, it cannot be complete. Remember, its Godel's Incompleteness Theorem,
>not Godel's Inconsistancy Theorem. A consistant system has no 
>contradicting statements. A complete system can prove all statements 
>expressable in that system. You seem to have mixed up the two in places.

Yes, a system that can prove its own consistency *is* complete,
because it is inconsistent. Godel's first and second incompleteness
theorems proved two things about axiomatic systems at least as
powerful as Peano arithmetic: (1) They are all either inconsistent or
incomplete, and (2) If a system is consistent, then it cannot prove
its consistency.

(2) is logically equivalent to: If a system can prove its own
consistency, then it is, in fact, inconsistent.

Daryl McCullough
ORA Corp.
Ithaca, NY

