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Article 5979 of comp.ai.philosophy:
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>From: torkel@sics.se (Torkel Franzen)
Newsgroups: comp.ai.philosophy
Subject: Re: penrose
Message-ID: <1992May29.130100.13926@sics.se>
Date: 29 May 92 13:01:00 GMT
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In-Reply-To: costello@CS.Stanford.EDU's message of 29 May 92 11:35:59 GMT

In article <1992May29.113559.14311@CSD-NewsHost.Stanford.EDU> costello@CS.
Stanford.EDU (T Costello) writes:


  >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.

  It might indeed if one doesn't take into account what is meant by a
recursive progression. Which was the point of my original comment.

>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.

  You are mangling the concepts involved. First, progressions are not
sequences: they do not associate theories with ordinals but with notations
for ordinals, and notations denoting the same ordinal may be assigned
different theories. Second, "autonomous" is not a property of theories, but
of recursive progressions, and I fail to see what your "theories where we
insist that we have this restriction" refers to. I don't understand your
further comments.


