Bruno,
              Thanks for this great refresher course.    
                                                                                
marty a.




  ----- Original Message ----- 
  From: Bruno Marchal 
  To: everything-list@googlegroups.com 
  Sent: Friday, March 19, 2010 5:59 AM
  Subject: Re: Free will: Wrong entry.




  Marty,




                   Can you clarify the origins of the Lobian Machine? Does it 
arise out of the theorem of Hugo Martin Lob? 




  Yes. I have often explained that theorem, years ago on this list (and 
elsewhere) and I can have opportunities to explain it again. You can see some 
of my papers where I explain it, including SANE2004.


  Löb's theorem is a generalization of Gödel's theorem. It is related to a 
funny proof of the existence of Santa Klauss, for those who remember.


  Löb's theorem is very weird. It says that Peano Arithmetic PA (and all Lobian 
entity) are close for the following inference rule. If the theory proves Bp -> 
p, then the theory proves p. It makes the theory (machine) modest: it proves Bp 
-> p, only when he proves p (in which case Bp -> p follows from elementary 
classical logic). PA can prove its own Löb's theorem, and this leads to the Löb 
formula: B(Bp -> p) -> Bp. And this *is* the (main) axiom of G and G*.
  (Bp = provable p, p some arithmetical proposition (or its gödel number when 
in the scope of "B").


  In particular the theory cannot prove Bf -> f   (f = constant false 
proposition), they would prove B(Bf->f), and by modus ponens and Löb's formula 
Bf, and by modus ponens again: f. Thus they cannot prove their own consistency 
(Bf -> f = ~Bf = ~~D~f = Dt). This is Gödel's second incompleteness theorem.


  Löb's discovery is a key event in the mathematical study of self-reference.




    Is it shorthand for the "lobes" of the human brain?


  No. :)






    What is the difference between a lobian machine and a universal lobian 
machine? And how do they relate to the question of free will? Many thanks,


  It happens that universal machines become Löbian (obey Löb's rule, and prove 
its formal version: Löb's formula) once they know (in some very weak technical 
sense) that they are universal.


  So you can just keep this in mind: a lobian machine is a universal machine 
which knows that she is universal. It obeys to the Löb's formula and indeed of 
the whole of G and G*. It has the arithmetical "Plotinian theology".


  Knowing that they are universal, they can study they own limitations, develop 
theologies (distinguishing proof and true), and develop free-will, from their 
own point of views. They can distinguish all the person-notions, the 8 
hypostases, etc. 


  They are also sort of "universal dissident", i.e. capable to refute any 
complete theory about them. They provide a tool for demolishing all 
reductionist interpretation of reductive comp theories. Some reduction are not 
reductionist.


  Their existence is responsible for the mess in Platonia: the impossibility to 
unify in one theory the whole arithmetical truth.


  Bruno






  http://iridia.ulb.ac.be/~marchal/







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