Le 12-août-07, à 18:00, John Mikes a écrit :
> 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 think you have missed the posts where I defined Angels, Gods,
Supergods, etc. By definition they refer to lobian entities which are
NOT emulable by Turing Machines. They exists mathematically. They are
the main object study of a branch of mathematical logic known as
recursion theory or computability theory (which could be called
uncomputability theory aswell).
A detailed example of a very powerful, yet lobian, "angel" is given in
Boolos 93, and called "Analysis + Omega-rule", and I have often refer
to it by calling it Anomega.
Perhaps later I will explain that the full (first order modal logical
system) which I use to interpret Plotinus "divine intellect" is really
an angel too, actually more powerful than the unnameable "god" (the
plotinus' ONE) of the machine.
> I read Kim's question as a joke, you took it seriously with some
> (imagined) meaning you had in mind. Faith?
I remind you that we have already talk a lot about the necessity of
some "faith" from the part of lobian entities (machine or not). The
machine cannot prove its own consistency, but can bet on it, and use
that bet in many different ways.
> Please, do not tell me that your theories are as well applicable to
> faith-items! Next time sopmebody will calculate the enthalpy of the
Don't worry. each term I am using have been well defined. By "Angel" I
just mean those lobian entities which are not machines. I did already,
in 2000, in this list called G* the "guardian angel" of the machine,
because it knows a lot about the machine that the machine cannot know
or prove about itself. Now, with the arithmetical interpretation of
Plotinus, I have to use those terms in a bit more systematic ways. The
G/G* type of theology works for (ideally correct) machine, but also on
many self-referentially correct entities which are NOT machine. OK?
> 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
>> verifiable (decidable) property". PI_1 sentences asserts something
>> "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)
>> of not being the Godel number (or any arithmetical encoding) of a
>> 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*,
>> 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
>> 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
>> P1-complete *in* Arithmetical Truth (making bigger than the
>> 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.
>> 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
>> 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
>> 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.
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