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From: franka@mmintl.UUCP (Frank Adams)
Newsgroups: net.math
Subject: Re: Relative Number Theory
Message-ID: <1148@mmintl.UUCP>
Date: Fri, 7-Feb-86 17:52:31 EST
Article-I.D.: mmintl.1148
Posted: Fri Feb  7 17:52:31 1986
Date-Received: Wed, 12-Feb-86 07:28:49 EST
References: <2314@umcp-cs.UUCP> <2232@pyuxd.UUCP> <2393@umcp-cs.UUCP> <1079@mmintl.UUCP> <4516@kestrel.ARPA>
Reply-To: franka@mmintl.UUCP (Frank Adams)
Organization: Multimate International, E. Hartford, CT
Lines: 25

In article <4516@kestrel.ARPA> ladkin@kestrel.ARPA writes:
>(adams)
>>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?  
>
>What is N(a)?

This was in the original posting, but got cut out when it was quoted.

N(0) is the formal system axiomitized by the Peano postulates.  For each
formal system N(a), P(a) is the Goedel sentence for the system.  For any
computable ordinal a, N(a) is the system whose axioms consist of the
Peano postulates, plus the statements P(b) for every ordinal b<a.
(In particular, N(a') has all the axioms of N(a), plus P(a).)

The requirement that a be a computable ordinal is required in order for
N(a) to be a formal system; if u is the smallest uncomputable ordinal,
it is not decidable whether a statement P is an axiom of N(u).  At that
point the Goedelization fails, so there is no statement P(u).

(I am going on vacation, and will not return until March.  If you want
me to see a response, mail it to me.)
Frank Adams                           ihnp4!philabs!pwa-b!mmintl!franka
Multimate International    52 Oakland Ave North    E. Hartford, CT 06108