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Article 5976 of comp.ai.philosophy:
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>From: costello@CS.Stanford.EDU (T Costello)
Newsgroups: comp.ai.philosophy
Subject: Re: penrose
Message-ID: <1992May29.112749.14160@CSD-NewsHost.Stanford.EDU>
Date: 29 May 92 11:27:49 GMT
Article-I.D.: CSD-News.1992May29.112749.14160
References: <atten.706786286@groucho.phil.ruu.nl> <1992May29.053625.6202@sics.se>
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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:
|> 
|>    >Your informal reflection principle causes the system to
|>    >be inconsistent.
|> 
|>   Look at this a bit more carefully. The axioms you formulate start from
|> apparently plausible principles for a supposed notion of "provable" and
|> lead to a contradiction. What I formulated was just a well understood
|> set of reflection axioms by which any sound theory may be soundly extended.
|> I suggest that you think about your axioms for a while and find out for
|> yourself why they don't apply to the concept "provable in T" for sound T.
|> 

I am afraid I did not make myself quite clear.  Given a first order language, take 
Robinson arithmetic, or a system at least as strong.  Take the
set of axioms and the inference rules.  For the language construct
a mapping from its symbols to numbers, and hence from numbers to equations.
Now consider the elementary predicate defined as follows.  It is true of
the encoding of each axiom, if it is true of an implication and it
first part it is true of its second. 


We add that this predicate is true of the encoding of the 
statement that any number it applies to is true.  This predicate exists,
call it B. It of course already has a aritmetical name. A model of this can be demonstrated.

Now consider the reflection principle you suggested, that is the connecting of
the predicate that states that the arithmetic coding of a proof of a statement
 implies a statement.  Our notion of proof is a proof from the axioms given above.
We use the same encoding scheme.  We can write this as

Bew x \supset x

But we see from our definition of B above that we can derive that Bx \supset Bew x.

Thus by Montague's result we have an inconsistent system.  

I applied your reflection principle to a consistent system to reach an inconsistent 
system.

Thus I claim that your reflection principle does not hold in general.

Tom


