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Article 5977 of comp.ai.philosophy:
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>From: costello@CS.Stanford.EDU (T Costello)
Newsgroups: comp.ai.philosophy
Subject: Re: penrose
Message-ID: <1992May29.113559.14311@CSD-NewsHost.Stanford.EDU>
Date: 29 May 92 11:35:59 GMT
References: <atten.706786286@groucho.phil.ruu.nl> <1992May29.053625.6202@sics.se>
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Organization: Computer Science Department, Stanford University
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In article <1992May29.053625.6202@sics.se>, torkel@sics.se (Torkel Franzen) writes:
|> In article <1992May29.012700.7102@CSD-NewsHost.Stanford.EDU> costello@CS.
|> Stanford.EDU (T Costello) writes:
|>   What strikes you as strange about the result?

	The strange result is of course that, while the notion of a recursive progression
	seems meaningful, we cannot have a autonomous progression, because of the non elementary
	nature of the predicate we would have to apply induction to to express this sequence.

	Tom

The above paragraph is a little difficult to understand.  What I meant was that 
it might be thought that the fact that we can get all the true statements of
arithmetic from a recursive progression might contradict Godel's theorem.

The reason it does not, is because in general we cannot prove that a given d for
a theory Ad in the progression is an ordinal in the sequence.  Theories
where we insist that we have this restriction are called autonomous theories,
and the completeness results do not hold for them.  The reason that we cannot
express these ordinals is because as in transfinite induction, we can only 
get as far as omega to the power of omega etc, repeated a finite number of times
for any given system.  The notion of it applied omega times would be got by
induction on a predicate.  However this predicate is non elementary, and thus we cannot
do indcution over it.

Tom


