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?

`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 universallobian 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/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.