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Article 5922 of comp.ai.philosophy:
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>From: costello@CS.Stanford.EDU (T Costello)
Newsgroups: comp.ai.philosophy
Subject: Re: penrose
Message-ID: <1992May26.225220.18126@CSD-NewsHost.Stanford.EDU>
Date: 26 May 92 22:52:20 GMT
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In article <1992May26.184457.4065@guinness.idbsu.edu>, holmes@opal.idbsu.edu (Randall Holmes) writes:
|> In article <atten.706874854@groucho.phil.ruu.nl> atten@phil.ruu.nl (Mark van Atten) writes:
|> [...]
|> >
|> >Penrose maintains that the rail-jumping cannot be formalized, because
|> >the construction of a new Goedel sentence cannot be formalized. See his
|> >article in Behavioral and Brain Sciences, 1991.
|> 
|> It is possible to formalize the process of constructing a new Godel
|> sentence in limited contexts.  For instance, there is a mechanical
|> procedure which, given a theory, will produce another theory which
|> proves its Godel sentence, and this procedure can then be applied
|> repeatedly.  Where does this break down?  The union of the sequence of
|> theories constructed by iterated application of this procedure has a
|> Godel sentence, too, which will not be provable in any of the theories
|> constructed by the procedure; on the other hand, we can design an even
|> smarter procedure which deals with this case, too (and has trouble
|> further on).  It is not at all clear that we are not somewhere in this
|> hierarchy of partial "rail-jumpers".  (It is intuitively apparent to
|> me that we _are_, but intuition is unreliable...)  
|> 
|> > >Best wishes,
|> >Mark.
|> 
|> 
|> -- 
|> The opinions expressed		|     --Sincerely,
|> above are not the "official"	|     M. Randall Holmes
|> opinions of any person		|     Math. Dept., Boise State Univ.
|> or institution.			|     holmes@opal.idbsu.edu


"Rail Jumping" does not break down when you reach a limit ordinal.  As
you said above you take the union of all the theories below it.  If we take as our notion 
    A_{k+1} consists of A_{k} together with all the sentences of the form
 \forall x Pr_{A_k}(\phibar(x)) \supset \forall x. \phi(x)

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.  

This is explained in detail in Feferman's "Transfinite recursive progressions of 
axiomatic theories" JSL 1962.   

Tom 


