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