Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site mmintl.UUCP
Path: utzoo!watmath!clyde!burl!ulysses!bellcore!decvax!genrad!panda!talcott!harvard!cmcl2!philabs!pwa-b!mmintl!franka
From: franka@mmintl.UUCP (Frank Adams)
Newsgroups: net.math
Subject: Re: Conditioning for Reason
Message-ID: <1079@mmintl.UUCP>
Date: Mon, 27-Jan-86 17:58:11 EST
Article-I.D.: mmintl.1079
Posted: Mon Jan 27 17:58:11 1986
Date-Received: Sat, 1-Feb-86 04:38:22 EST
References: <2314@umcp-cs.UUCP> <2232@pyuxd.UUCP>
Reply-To: franka@mmintl.UUCP (Frank Adams)
Organization: Multimate International, E. Hartford, CT
Lines: 21

In article <11480@ucbvax.BERKELEY.EDU> weemba@brahms.UUCP (Matthew P. Wiener) writes:
>>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.
>
>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.

Can you prove that there is such a problem which is not solvable in N(a)
for any constructable ordinal a?  It seems plausible, but plausibility and
proof are two different things.

Frank Adams                           ihpn4!philabs!pwa-b!mmintl!franka
Multimate International    52 Oakland Ave North    E. Hartford, CT 06108