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From: jeff@aiai.ed.ac.uk (Jeff Dalton)
Subject: Re: Penrose's new book
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References: <1994Oct19.132318.18033@oracorp.com>
Date: Sat, 5 Nov 1994 01:10:30 GMT
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In article <1994Oct19.132318.18033@oracorp.com> daryl@oracorp.com (Daryl McCullough) writes:
>jeff@aiai.ed.ac.uk (Jeff Dalton) writes:
>
>>> daryl@oracorp.com (Daryl McCullough) writes:

>>>     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.

So why _were_ you saying "if we are *certain*" above?

>>Here's another data point for you: there are proofs of the consistency
>>of arithmetic.
>
>Not using arithmetic. 

It's true that it's not a proof using only the system it's proving
the consistency of.

-- jd
