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From: ladkin@kestrel.ARPA
Newsgroups: net.math
Subject: Re: Relative Number Theory
Message-ID: <4666@kestrel.ARPA>
Date: Fri, 7-Feb-86 20:02:47 EST
Article-I.D.: kestrel.4666
Posted: Fri Feb  7 20:02:47 1986
Date-Received: Tue, 11-Feb-86 03:57:26 EST
References: <2314@umcp-cs.UUCP> <2232@pyuxd.UUCP> <2393@umcp-cs.UUCP> <11727@ucbvax.BERKELEY.EDU>
Organization: Kestrel Institute, Palo Alto, CA
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In article <11727@ucbvax.BERKELEY.EDU>, weemba@brahms.BERKELEY.EDU (Matthew P. Wiener) writes:
> 
> My guess is that N(a) is Peano Arithmetic + induction on well-orderings of
> type less than a.  The proof strength does grow (like a step function) as
> you increase a among the recursive ordinals.  For example, if e=epsilon_0,
> ie, the limit of w,w^w,w^w^w,w^w^w^w,... (where w=omega), then N(e) is, by
> Gentzen's theorem, able to prove the consistency of PA.
> 
That sounds reasonable, but notice Adams wanted N(a) for ANY 
ordinal a. Your interpretation doesn't address 
0mega-1 Church Kleene and above, since these well-orders 
aren't in N, and have to be added somehow.

Peter Ladkin