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 & 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...".


(*) with the notable and fundamental exception of the pure first person knower.


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