# Re: Free will: Wrong entry.

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Marty,```
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Can you clarify the origins of the Lobian Machine? Does it arise out of the theorem of Hugo Martin Lob?
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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.
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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.
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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").
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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.
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Löb's discovery is a key event in the mathematical study of self- reference.
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```Is it shorthand for the "lobes" of the human brain?
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No. :)

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

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

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