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From: weemba@brahms.BERKELEY.EDU (Matthew P. Wiener)
Newsgroups: net.math
Subject: Re: Conditioning for Reason
Message-ID: <11480@ucbvax.BERKELEY.EDU>
Date: Tue, 21-Jan-86 01:52:22 EST
Article-I.D.: ucbvax.11480
Posted: Tue Jan 21 01:52:22 1986
Date-Received: Thu, 23-Jan-86 08:24:33 EST
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Reply-To: weemba@brahms.UUCP (Matthew P. Wiener)
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>The question I don't know the answer to is the following: is there any
>closed number-theoretic formula X, such that neither X nor not X is provable
>in the system N(a) for any computable ordinal a?  (I would guess that
>the question is not decidable.)  If there is such an X, then it may be
>correct to assert that there are statements of number theory which are
>not accessable to reason.  If there is no such X, then the assertion is
>false.  (The commonly quoted conclusion that there are true statements
>which we cannot know is based on the assumption that we are in fact
>equivalent to a Turing machine, and thus limited to the theory N(a) for
>some ordinal a < u.  In other words, "not accessible to *our* reason",
>not "not accessible to reason".)

Frank, I think you do know the answer, but got too carried away by the
discussion!  The consistency of set theories, either stronger or weaker
than ZFC, can all be coded into questions about whether or not a certain
polynomial has integral solutions, by the Matiyasevich theorem.  These
Diaphontine problems should answer your question.

For further details, see my recent posting in net.{philosophy|ai},
"Re: a halting problem: a meaty response", where a similar question
came up with about the same answer.

ucbvax!brahms!weemba	Matthew P Wiener/UCB Math Dept/Berkeley CA 94720