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Article 5938 of comp.ai.philosophy:
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>From: costello@CS.Stanford.EDU (T Costello)
Subject: Re: penrose
Message-ID: <1992May27.105042.29890@CSD-NewsHost.Stanford.EDU>
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Organization: Computer Science Department, Stanford University
References: <atten.706786286@groucho.phil.ruu.nl> <1992May27.080114.8344@sics.se>
Date: Wed, 27 May 1992 10:50:42 GMT
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In article <1992May27.080114.8344@sics.se>, torkel@sics.se (Torkel Franzen) writes:
|> In article <1992May26.225220.18126@CSD-NewsHost.Stanford.EDU> 
|> costello@CS.Stanford.EDU (T Costello) writes:
|> 
|>    >The set of theorems of the above progression includes all true sentences of
|>    >elementary number theory.  The progression through the ordinals has order
|>    >type less than omega one.
|> 
|>   You haven't, of course, introduced the very technical concept of
|> "progression" used in Feferman's paper, and your remarks may be misleading.
|> It should be emphasized that there is nothing in this work that implies
|> that the set of theorems "humanly provable" by reflection is not
|> recursively enumerable.
|> 
|>   What is epistemologically interesting and (by Godel's theorem) at
|> least potentially useful is the informal reflection principle
|> "if a theory T is sound, any extension by reflection of T is also sound".
|> This is not a formal principle, but there is nothing about it that
|> suggests that it should be peculiarly unimplementable on a machine.

I must disagree.  The principle you have stated can be formalised as follows.

Let Bx stand for there is a proof of x.

Then take theses four axioms in addition to the logical tautologies.
Bx \supset x
B ( B x \supset x)
B(axiom) where axiom is a logical axiom
B(x) \land B( x \supset y) \supset By

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.  And thus shows that your notion
is inconsistent.  I f you disagree with one of these axioms I would be most
interested in which you feel is incorrect, or does not capture what you
mean by reflecting. 

The basic result above is from Montague, A Paradox Regained.

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.

Tom


