From newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.ecf!utgpu!torn.onet.on.ca!watserv1!watmath!xenitec!uunet.ca!uunet!pmafire!mica.inel.gov!guinness!opal.idbsu.edu!holmes Sun May 31 19:04:13 EDT 1992
Article 5915 of comp.ai.philosophy:
Newsgroups: comp.ai.philosophy
Path: newshub.ccs.yorku.ca!ists!helios.physics.utoronto.ca!news-server.ecf!utgpu!torn.onet.on.ca!watserv1!watmath!xenitec!uunet.ca!uunet!pmafire!mica.inel.gov!guinness!opal.idbsu.edu!holmes
>From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: Re: penrose
Message-ID: <1992May26.184457.4065@guinness.idbsu.edu>
Sender: usenet@guinness.idbsu.edu (Usenet News mail)
Nntp-Posting-Host: opal
Organization: Boise State University Math Dept.
References: <atten.706786286@groucho.phil.ruu.nl> <1992May25.164314.19628@guinness.idbsu.edu> <atten.706874854@groucho.phil.ruu.nl>
Date: Tue, 26 May 1992 18:44:57 GMT
Lines: 29

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


