You are right when it comes to the combination of two independent
systems A and B. What the original poster's idea was a
self-simulating, or self-aware system. In this case, consider the liar
type paradox:

  I cannot prove this statement

Whilst I cannot prove this statement, I do know it is true, simply
because if I could prove the statement it would be false.

To know that it is true, I am using self-reference about my own proof
capabilities. 

I don't think anyone yet has managed a self aware formal system,
although self-reproducing systems have been known since the 1950s, and
are popularly encountered in the form of computer viruses. There has
to be some relationship between a self-reproducing system and a
self-aware system...

Cheers

On Thu, May 24, 2007 at 09:45:45PM -0400, Jesse Mazer wrote:
> 
> I definitely don't think the two systems could be complete, since (handwavey 
> argument follows) if you have two theorem-proving algorithms A and B, it's 
> trivial to just create a new algorithm that prints out the theorems that 
> either A or B could print out, and incompleteness should apply to this too.
> 
> Jesse
> 
> 
> >From: Russell Standish <[EMAIL PROTECTED]>
> >Reply-To: [EMAIL PROTECTED]
> >To: [EMAIL PROTECTED]
> >Subject: Re: Overcoming Incompleteness
> >Date: Thu, 24 May 2007 23:59:23 +1000
> >
> >
> >Sounds plausible that self-aware systems can manage this. I'd like to
> >see this done as a formal system though, as I have a natural mistrust
> >of handwaving arguments!
> >
> >On Thu, May 24, 2007 at 10:32:29AM -0700, Mohsen Ravanbakhsh wrote:
> > > Thanks for your patience! , I know that my arguments are somehow
> > > raw and immature in your view, but I'm just at the beginning.
> > >
> > > *S1 can simulate S2, but S1 has no reason to believe whatever S2 says.
> > > There is no problem.
> > > **Hofstadter "strange loop" are more related to arithmetical
> > > self-reference or general fixed point of recursive operator*
> > >
> > > OK then it, becomes my own idea!
> > > Suppose S1 and S2 are the same systems, and both KNOW that the other one 
> >is
> > > a similar system. Then both have the reason to believe in each others
> > > statements, with the improvement that the new system is COMPLETE. We've 
> >not
> > > exploited any more powerful system to overcome the incompleteness in our
> > > system.
> > > I think this is a great achievement!
> > > It's actually like this: YOU believe in ME. THEY give
> > > you a godelian statement (You theoretically can not prove this
> > > statement) you give it to ME and then see that I can neither prove it
> > > nor disprove it, so you tell
> > > THEM that their statement is true.
> > > But the wonder is in what we do just by ourselves. We have a THEORY OF 
> >MIND.
> > > You actually do not need to ask me about the truth of that statement, 
> >you
> > > just simulate me and that's why I can see the a godelian statement is at
> > > last
> > > true. But in the logical sense ONE system wont be able to overcome the
> > > incompleteness,
> > > so I might conclude:
> > > I'M NOT ONE LOGICAL SYSTEM!
> > > This is how we might rich a theory of self. A loopy(!) and multi(!) 
> >self.
> > >
> > >
> > >
> > > *
> > >
> > > *Mohsen Ravanbakhsh
> > >
> > > >
> >
> >--
> >
> >----------------------------------------------------------------------------
> >A/Prof Russell Standish                  Phone 0425 253119 (mobile)
> >Mathematics
> >UNSW SYDNEY 2052                      [EMAIL PROTECTED]
> >Australia                                http://www.hpcoders.com.au
> >----------------------------------------------------------------------------
> >
> >>
> 
> _________________________________________________________________
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> 
-- 

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A/Prof Russell Standish                  Phone 0425 253119 (mobile)
Mathematics                              
UNSW SYDNEY 2052                         [EMAIL PROTECTED]
Australia                                http://www.hpcoders.com.au
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