Hello Stephen,

On 23 Feb 2011, at 20:39, Stephen Paul King wrote:

Dear Bruno, Could you explain this a bit more? “The ideally correct machine is to the human what a material point is to the sun. My answer tries only to help you to understand what I mean by a knowing machine, not really a knowing human. Human have non-monotonic layers, they can update beliefs. The logic G and G*, and the six intensional variants can be seen as the tangential theology, but we are variable machine. G and G* remains invariant, but get each nanosecond (say) a different arithmetical interpretations. To extract physics, you need only the self-referential invariant.”How does one define a self-referential invariant? What might bean example of this?

`G, G*, all the arithmetical points of view (hypostases), are such`

`invariant.`

`It is a theorem that all the sound recursively enumerable extensions`

`of PA have the modal logics G and G* describing (completely at the`

`propositional level) their probability and consistency predicates. G`

`describe what they can prove, and G* what is true about them,`

`including what they cannot prove (like their consistency Dt = ~B~f).`

All this is a consequence of

`1) the second recursion theorem of Kleene, which you can see as a`

`fixed point theorem.`

`2) Gödel's diagonalization lemma: which really is the fact that PA`

`(and extensions) proves the Kleene recursion theorem themselves.`

`G and G* remain correct for the supermachines (machine with oracles),`

`and are even still correct and complete for many interesting`

`"gods" (non-turing emulable entities). G and G* remain correct but no`

`more complete, when getting very close to "God" (arithmetical truth).`

`I still don't know if they make sense for some truth theory. I doubt so.`

Bruno PS I recall (for the new people) that G is the modal logic having B(p->q) -> (Bp -> Bq) (Kripke's formula) B(Bp -> p) -> Bp (Löb's formula)

`as axioms, together with classical propositional calculus, and with`

`the usual modus ponens rule, and the necessitation rule (from p deduce`

`Bp) as deductive inference rule.`

`G* has as axioms the theorem of G, + the formula Bp -> p, but is NOT`

`close for the necessitation rule.`

`See the book by Boolos for more, or the archive (search 'S4Grz',`

`perhaps, or read the second half the sane04 paper).`

Best Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.