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From: jeff@aiai.ed.ac.uk (Jeff Dalton)
Subject: Re: Penrose's new book
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Date: Mon, 17 Oct 1994 18:17:10 GMT
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In article <1994Oct11.200236.25598@oracorp.com> 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.

But the details matter.  Is this summary a direct quote?

>     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*..."?

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

-- jeff



