From: Geoffrey Norman Watson 
Date: Tue, 26 Nov 1996 11:32:21 +1000
Organization: Computer Science Dept, University of Queensland


On Fri, 22 Nov 1996, Jim Balter wrote:

> Phil Roberts, Jr. wrote:
> > 
> > On Wed, 13 Nov 1996 22:03:17 GMT
> > Jim Balter wrote:
> > >
> > >In article <328A2BEC.70E1@ix.netcom.com>,
> > >Phil Roberts, Jr.  wrote:
> > 
> > >>You're right.  Lucas is _trivially_ wrong, in that he referred to his
> > >>attempt to extend Godel to the material realm as a _proof_ of the falsity
> > >>of mechanism (last paragraph).  What he offered was merely an argument,
> > >>but a pretty damn good one, if I'm not mistaken:
> > >
> > >You are very mistaken.
> > >
> > >>      Godel's theorem states that in any consistent system
> > >>      which is strong enough to produce simple arithmetic
> > >>      there are formulae which cannot be proved-in-the-system,
> > >>      but which we [standing outside the system] can _see_ to
> > >>      be true.
> > >
> > 
> > Its important to understand that this is not Lucas's idiosyncratic view of
> > Godel.  This _is_ what Godel's theorem amounts to when translated into
> > simple English.
> 
> Godel's theorem does not refer to "we", thus this must be false.
> 
> I, for one, have never seen to be true the Godel sentence of any system;
> in fact I've never seen such a sentence.  Have you?  Can you state one?
> 
> If I did see one, I certainly wouldn't just "see" that it is true,
> but would have to have that proven to me.  Of course, the proof
> could not be given in the system for which the sentence is the
> Godel sentence.
> 
 This whole debate seems to be getting a little confusing.  

 One point:
 Goedel's theorems are about axiomatic systems, and the main result of his 
 work was that axiomatic systems are not capable of supporting mathematics 
 directly, and so demolished Hilbert's program. Of course, most of mathematics
 is done using rigorous rather than formal argument.
 
 Note that Goedel's first incompleteness theorem showed that arithmetic is
 not axiomatisable. 

 A second point: 
 An important aspect of these results is that in some way "provability" is 
 weaker than "truth" (this has the advantage that provability thus escapes 
 from the truth paradoxes) and they explore the differences.  So it is wrong
 to equate the two by saying "it can't be true unless you can prove it".  
 (Although of course with a consistent system, then we can say "if you can 
  prove it then it is true".)

 I believe that detailed versions of Goedel's second incompleteness theorem
 exhibit quite specifically the sentences which are true but which cannot
 be proved in the relevant system. Their truth is established by the law of
 the excluded middle and the meta-level relationship between statements 
 of arithmetic and sentences of the theory defined by the Goedel encoding.
  
 A theory may have many interpretations, and it may well be that we can 
 recognise truths in some interpretations that cannot be proved in the theory.

-----------------------------------------------------------------------------
  Geoffrey Watson                                    gwat@cs.uq.edu.au