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 <marchal.domain.name.hidden> 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/
>
>
> >
>
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Received on Sun Aug 12 2007 - 12:00:54 PDT