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From: "Raymond J Nawara Jr." <dreck@rahul.net>
Subject: Re: Penrose's new book
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Date: Mon, 24 Oct 1994 19:21:53 GMT
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Daryl McCullough (daryl@oracorp.com) wrote:

: It doesn't. Here's the extended version of Godel's theorem:

:      Let L be any countable language, and let T be any theory
:      (collection of statements closed under logical deduction) in
:      the language of L. Assume that T is "sufficiently self-referential"
:      in the sense that (1) there is a way to code up expressions in
:      L as terms of L such that all computable operations on expressions
:      are definable, and (2) the property of "being the code of a theorem
:      of T" is expressible. Then there is a statement con(T) in the language
:      of T coding the claim that T is consistent, and if T is consistent,
:      then T cannot prove con(T).

: It is not necessary that the theory T be computable for an incompleteness
: theorem to apply.

: Now if you let the language L be full English, and let T be the
: collection of statements that Penrose can eventually come to be
: certain are true, then conditions (1) and (2) hold. So, if the
: collection T is consistent, then it cannot contain the formalization
: of the claim that T is consistent.

: >Here's another data point for you: there are proofs of the consistency
: >of arithmetic.

: Not using arithmetic. Let me show how the extended Godel's theorem
: works in a few cases:

: 1. Peano arithmetic: Peano arithmetic is sufficiently self-referential
: that the property of "being the code of a theorem of Peano Arithmetic"
: is expressible. Therefore, there is a statement coding the claim that
: PA is consistent, and PA cannot prove that statement.

: 2. "True" arithmetic: By one of Tarski's theorems, there is no way to
: formalize theoremhood in arithmetic in the language of arithmetic. So
: condition (2) in the above statement of the extended Godel's theorem
: doesn't hold. True arithmetic cannot prove its own consistency,
: because it can't even *express* the claim that it is consistent.

: 3. ZF (set theory) + true arithmetic (as formalized in ZF): this theory
: is more powerful than true arithmetic, but once again, the extended
: Godel's theorem applies. There is a statement expressing the claim
: that this theory is consistent, and this statement is not provable in
: the theory.


: 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.

Not having read Penrose, just this thread, I still think the statement that
the human mind is consistant is the flaw in the argument. 




-- 
-- Dreck
