Le 13-janv.-06, à 02:24, Russell Standish a écrit :
I have tried to identify 1pp with G and 1p with G*, but I'm really
unsure that the analogy is sound.
It is tempting to classify the self-reference logics (G, G*) in the
first person discourses, and I have been stuck in that idea for a
while.
But it can't be. The godelian provability predicate is definable in
arithmetic only thanks to the fact that it is just asked to Peano
Aritmetic (a "famous" lobian or self-referentially correct machine) to
talk about a third person description of itself, through Godel numbers
usually. So G is more like the discourse you could do on your
doppelganger or on your brain, seen in some high level third person
description. It is a purely scientific (and even purely deductive)
third person talk a machine can do about itself, except that she did
already bet on some level of substitution, so that in a sense it is
just serendipitously correct.
Now if the lobian machine we talk with is sufficiently elementary, like
Peano Arithmetic, and most probably Zermelo Fraenkel Set Theory, we can
believe in their consistency (~Bf) and even in their soundness (Bp ->
p): PA (resp. ZF) does not tell us falsities (~Bf), and PA communicates
only true sentences of Arithmetic (Bp -> p)(resp. set theory).
Exercise: find a lobian machine which is unsound, but still consistent.
By "exercise" I mean I have not the time to explain, but I want draw
the attention on that important fact. It is a rather simple consequence
of the second incompleteness theorem: Dt -> ~BDt.
Amazingly enough perhaps, G* gives still a pure third person discourse.
Note in passing that I insist on the fact that G and G* gives thrid
persons discourses in my last three english paper.
G* is a proper extension of G, which is complete, at the propositional
level, for the description of the true provability and consistency
sentences. As applied to us, we cannot take it for granted because we
cannot know our correct level of description, and then, if we are lucky
enough to bet on the correct level, we still cannot know if comp is
true; and then, even if comp is true, if we are lobian machines, we
cannot know we are consistent, still less sound.
And this is actually what G* says. And so G* is an amazing sort of
"scientific theology" which ask you not to take it as granted in its
roots. Practically it means theotechnologies are private matters,
somehow, and it makes obligatory the right of saying no, to the doctor.
All the first persons notion are obtained by variant of the
Theaetetical definition of knowledge. This is possible exactly thanks
to the gap between provability (G) and truth (G*).
Although the following
Bp
Bp & p (pure first person, the knower)
Bp & Dp (first person plural, betting machines)
Bp & Dp & p
are shown equivalent by G*, and so defined strictly speaking the same
machine, none of those equivalencies can be shown by G, i.e. the
machine itself. So, although they define the same machine, from the
machine stance they defines different logics.
And both G and G* can derive completely (at the propositional level)
those logics. And of course here too G* knows more(*), so that the
theaetetical variants are lifted to the corona G* \ G.
That's the beginning of the story, it is the framework for translating
the UDA thought experiment in the language of the machine, and extract
the logics of "probability one" on the near 2^aleph_0
"observer-moments/states/worlds...".
Bruno
(*) with the notable and fundamental exception of the pure first person
knower.
http://iridia.ulb.ac.be/~marchal/
