On 18 Jun 2014, at 07:23, meekerdb wrote:
Bruno, I wonder if you're aware of this critique of Maudlin's
Olympia argument, which of course also applies to the MGA?
http://www.colinklein.org/papers/OlympiaOMachines.pdf
Hmm..... I should read that at ease, and not after ten hours of oral
exams, ... but why did he say that Olympia is a Turing Machine?
Only Olympia + the Claras incarnate a Turing machine.
The fact that Olympia and Olympia+the Claras are *physically*
equivalent (at the relevant level) is a problem for the physical
supervenience, but not necessarily for other form of supervenience
(like the comp one where consciousness is associated to all
computations going through the relevant states).
Hum... ( I read a bit) ... Not sure his definition of 'trurl' make
sense, nor his use of the "oracle". (by the way, I love Stanislam Lem)
Hmm... he is wrong on the notion of oracle.
An oracle is just a sort of divine entity complete for non computable
set.
A typical example is the "Halting oracle", which is pi_1 complete
(more powerful than a universal machine, it knows about the Riemann
Hypothesis!).
The tought was that with some such oracle, may be we could solve all
arithmetic problem. But such "gods" obeys the same incompletness
phenomenon, and a pi_1 god, for example, can still not filtrated the
total and strictly partial code/machines, in a universal enumeration
of machines/programs. For that you need a pi_2 god.
universal machine: complete for the provability of sigma_1 sentences,
type ExP(x) with P decidable/sigma_0.
After that you have the pi_1 complete sets, or notion (like
consistency, Halting, ..) which are thge non computable negation of
the sigma_1 sentences: ~ExP(x) = Ax ~P(x) and if P is sigma_0, ~P is
also sigma_0, and so the pi_1 formula are the formula of type AxP(x),
with P decidable.
Then you have the sigma_2 notions, ExAyP(x,y) the pi_2 notions (set,
relations), again, negation of the sigma_2 (using again the fact that
if P(x, y) is decidable, then ~P(x, y) is decidable too), so they are
equivalent of proposition of type AxEyP(x,y).
Oracles have been invented to explains that even when you might know
the whole pi_7 truth, you can't solve the sigma_8 problems.
Other type of oracles are possible, like the random oracle, which does
not add any power in the sense above, but can add a lot of power on
the tractability issue.
Is there a god having the knowledge of all the pi_i and sigma_i gods?
Of course, that's the definition of the arithmetical truth. But
machines cannot give it a name, and the set of all true proposition is
not definable in arithmetic.
And do the god "arithmetical truth" know everything?
No. For example, the quantified modal logics of provabilty qG and qG*,
are undecidable (unlike the propositional G and G*)), and indeed as
much as possible from their definitions. qG is pi_2 complete, and qG*
is pi_1 complete in the oracle of arithmetical truth.
In the comp theology, even with the full help of God, you have still
an infinite task to accomplish to get the Noùs/intelligible reality/
worlds of ideas.
Note that QM implies that nature provides a random oracle, as far as
we can "purify" some 1/sqrt(2)(0+1) state. For many tasks, classical
pseudo-random are enough.
Bruno
Brent
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