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
(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
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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