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From: ladkin@kestrel.ARPA
Newsgroups: net.math
Subject: Relative Number Theory
Message-ID: <4516@kestrel.ARPA>
Date: Mon, 3-Feb-86 15:42:34 EST
Article-I.D.: kestrel.4516
Posted: Mon Feb  3 15:42:34 1986
Date-Received: Wed, 5-Feb-86 01:43:11 EST
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Organization: Kestrel Institute, Palo Alto, CA
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(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)? It can't be Peano Arithmetic relativised to a,
since the induction axiom is false if a isn't omega. There
are non-standard models of PA but they're not well-founded,
and a is supposed to be an ordinal. It is also no longer
true that every number except 0 has a predecessor, when there
are limit ordinals in the domain.
If the ordinal's computable, then there's a notation for it
embedded in PA ( a recursive function coding the well-order),
which would mean that, whatever N(a) might be, it seems to
have the same proof theory as N ?
I'm running out of guesses.

Peter Ladkin