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 be an 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/



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