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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