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Article 5939 of comp.ai.philosophy:
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>From: torkel@sics.se (Torkel Franzen)
Subject: Re: penrose
In-Reply-To: costello@CS.Stanford.EDU's message of Wed, 27 May 1992 10:50:42 GMT
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Date: Wed, 27 May 1992 11:58:43 GMT
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In article <1992May27.105042.29890@CSD-NewsHost.Stanford.EDU> costello@CS.
Stanford.EDU (T Costello) writes:

   >The above is inconsistent if ajoined to a system at least as strong as
   >Robinson arithemetic.  I believe it accurately characterises the idea
   >of reflection that you descibed earlier.

  Your axioms have nothing in particular to do with the informal
reflection principle I formulated. To repeat, that principle was
"if a theory T is sound, any extension by reflection of T is also
sound". Here "extension by reflection" is itself not a formal concept,
but refers to various reflection axioms that may be added. The basic
formal reflection principle by which any sound theory T may be soundly
extended consists in the axiom schema

          BevT(+A+) -> A

for every sentence A, where +A+ is a name of the statement A and BevT a
suitably defined provability predicate for T. I don't pretend to understand
any notion "A is provable", where "provable" does not mean provable in a
specified theory.

   >The notion of progression is not that complicated in Fefermann's paper.
   >It merely consists of adding axioms for each ordinal.  The system itself
   >acts as one might suspect.  The strangeness of the result ( in the light
   >of Godel) does not come from any magic in the progression.

  Your description is inadequate. Recursive progressions are much more
complicated than transfinite sequences of theories, although the
latter are rather more interesting from a philosophical point of view.
Feferman notes that "with regard to questions of completeness for
transfinite sequences of theories, these results must be treated with
care. Because of the intensional character of the construction of
recursive progressions [the theories associated with ordinal notations
d and d' may be different, even though d and d' denote the same
ordinal]."

  What strikes you as strange about the result?


