Dear Bruno, did your scientific emotion just trapped you into showing that your theoretical setup makes no sense? Angels have NO rational meaning, they are phantsms of a (fairy?)tale and if your math-formulation can be applied to a (really) meaningless phantasy-object, the credibility of it suffers. How can your formalism be applied to something nonexistent? What does it say about the 'real' value of it?
I read Kim's question as a joke, you took it seriously with some (imagined) meaning you had in mind. Faith? Please, do not tell me that your theories are as well applicable to faith-items! Next time sopmebody will calculate the enthalpy of the resurrection. John On 8/9/07, Bruno Marchal <[EMAIL PROTECTED]> wrote: > > > > Le 09-août-07, à 11:22, Kim Jones a écrit : > > > > > What is "lobian" apart from la machine, Bruno? Are you referring to > > "angels" here? > > > > Aren't angels machines too? > > > Angels are not machine. Unless you extend the meaning of machine > 'course, but Angels' provability extend the provability of any > turing-emulable machine. Sometimes people use the term "supermachine" > for what I call angel, but mathematically, in principle, angels have > nothing to do with machine. Angels can prove any sentence having the > shape AxP(x) with P(x) decidable. (AxP(x) = For all x P(x)). Universal > machine are Sigma_1 complete. Angels are PI_1 complete. A sigma_1 > sentence asserts something like "It exists a number having such or such > verifiable (decidable) property". PI_1 sentences asserts something like > "all numbers have such or such verifiable (decidable) property". > The most famous PI_1 sentences is the *machine* consistency statement: > it is indeed equivalent with: all number have the (verifiable) property > of not being the Godel number (or any arithmetical encoding) of a proof > of f. > (f = any arithmetical contradiction, like (1+1=2 & ~(1+1=2)). > Angels can be shown to be lobian. They obey G and G*, and G and G* > describe completely their propositional provability logic. > (btw, I call "god" any non turing emulable entity obeying G and G*, but > for which G and G* are not complete (you need more axioms to > characterize their provability power; and I call supergods, entities > extending vastly the gods. > All that is really the subject matter of recursion theory, alias > computability theory (which should have been called, like someone said > in Siena, the theory of un-computability). recursion theory is really > the science of Angels and Gods, well before being the science of > Machines. But (and this is a consequence of incompleteness), you cannot > seriously study machines without studying angels too .... For example > the quantifies version of G* (the first order modal logic of > provability, the one I note qG*) can be shown to be a superangel: it is > P1-complete *in* Arithmetical Truth (making bigger than the "unnameable > God of the machine!!!!). This means that the divine intellect, or the > Plato's "NOUS" is bigger, in some sense than "God" (Plotinus' ONE). > Plato would have appreciate, and perhaps Plotinus too because he wants > the ONE to be simple ...., but yes the divine intellect is much more > powerful than the "God" (accepting the arithmetical interpretation of > the hypostases: see my Plotinus papert). > > I will certainly come back on all definitions. But roughly speaking, a > machine is (Turing)-universal (Sigma_1 complete) if it proves all true > Sigma_1 sentences. A machine is lobian if not only the machine proves > all true Sigma_1 sentences, but actually proves, for each Sigma_1 > sentence, that if that sentence is true then she can prove it. Put in > another way, a machine is universal if, for any Sigma_1 sentence S, it > is true that S->BS (B = beweisbar, provable). A machine is lobian if > she proves, for any Sigma_1 sentence S, S->BS. For a universal machine > (talking a bit of classical logic) S->BS is true about the machine. For > a lobian machine S->BS is not only true, but provable (again with S > representing Sigma_1 sentence). > > But all this is a theorem. My "abstract" definition of lobianity is: > any entity proving B(Bp->p)->Bp where B is her provability predicate. > A machine is weakly lobian if B(Bp->p)->Bp is true about the machine > (not necessarily provable). A typical weakly lobian system which is not > lobian is the modal logic K, I have talk about sometimes ago. > B(Bp->p)->Bp is the Lob formula (Loeb, or better Löb; better if well > printed!). > > Don't panic with all that vocabulary and formula, I will try, perhaps > with the help of people in the list, like David (if everything goes > well), to be more systematic. Please, indulge the fact that I could > change a definition in the course of the explanation, for a matter of > making things easier. > > But of course, ask any question, even if I decide to postpone the > comment, it can help me to figure out where are the difficulties. > > > 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 [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
