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From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Penrose's new book
Message-ID: <1994Oct19.132318.18033@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Wed, 19 Oct 1994 13:23:18 GMT
Lines: 103

jeff@aiai.ed.ac.uk (Jeff Dalton) writes:

>> daryl@oracorp.com (Daryl McCullough) writes:
>>
>>In Penrose' new book, he simply repeats the same arguments, together
>>with rebuttals to the criticism his first book received. Basically,
>>Penrose' argument is as follows:
>>
>>     1. Assume that our reasoning process is described by a formal
>>        theory F.
>>     2. Obviously, we would know that F is consistent, since it
>>        formalizes us and we know *we* are consistent.
>
>I'm not sure what you're getting at here, but it's pretty close
>to saying we're a model of F. If so, that looks like a pretty
>good way to show F is consistent.

Let me be a little more precise: change 1. to

       1'. Assume that there is an computable theory F in the language of
           arithmetic such that for every arithmetical formula A,
           F proves A if and only if human mathematical reasoning
           can come to know the truth of A.

That is the sense in which F describes our reasoning. Penrose claims
to show that something like 1' leads to a contradiction (or at least
to an implausibility).

>>     3. But by Godel's theorem, if F is consistent, then F cannot prove
>>        that F is consistent.
>>     4. Therefore, the statement "F is consistent" is something that
>>        we know, but F cannot prove.
>>     5. But that proves that F *doesn't* completely describe our
>>        reasoning process. Contradiction!
>>
>>The big hole in this argument is step 2. We have no reason to know
>>that our reasoning process is consistent. As a matter of fact, regardless
>>of whether or not our reasoning is computational, if we are *certain* that
>>our reasoning is consistent, then it isn't consistent. Penrose mistakenly
>>thinks that Godel's incompleteness theorem only applies to computational
>>systems. It applies to *any* set of statements in a language capable
>>of sufficient self-reference...including the set of statements believed
>>by Roger Penrose. If Penrose is convinced that his own reasoning process
>>is obviously free of contradictions, then he is just wrong.
>
>Since when does Godel's theorem say "if we are *certain*..."?

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."

Daryl McCullough
ORA Corp.
Ithaca, NY



