Le 13-janv.-06, à 02:24, Russell Standish a écrit :

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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/