On 20 Apr 2011, at 01:15, Stephen Paul King wrote:
You mentioned in a previous mail “the duality between Bp and Bp
&p”. Could you elaborate on this? Is it a Stone or a Pontryagin
duality? (these are different!)
Not at all. It is more akin to the cartesian mind-body duality. Bp
gives the third person discourse given by the machine, and Bp & p
gives the first person knower linked to that machine. Bp is third
person self-reference, like when you discuss with your doctor about
your temperature and body constitution, and Bp & p is the Plotinus
inner god, or soul, which the amchine cannot even name.
Also, are there any restriction on the content of the
p is for any arithmetical proposition (assuming comp)
p is for any sigma_1 proposition (assuming comp, and assuming that the
machine assumes comp too). In that case the hypostases are restricted
to the propositions accessed by the Universal Dovetailer.
Could a model of a possible world be a p?
No. p are (roughly speaking) finite syntactical object. A model is an
infinite structure. p might be satisfied, or not, by a model (true in
In the modal context, and in a model, we can use a stone-like duality
to "model" a proposition by a set of models. This is useful for the
hypostase which have no Kripke semantics. But this is technic. In Bp &
p, it is better to think of p as a syntactical arithmetical
proposition (it can be a huge one!).
I ask this because so far you seem to only consider p that are
tautologically true (such as 2+2 = 4)
p can be 2+2=5. In that case, the correct machine will not know p.
and thus are trivially independent of observer notions.
The observer is defined in the third person way by its body/program or
relative number (relative to some probable universal environement/
What about the contents of Observer Moments? Could they be p?
They can be accessible "p", like "I am in Washington" after a self-
duplication experience, or in the universal dovetailing.
I suspect that the in the “&p” (and p is true) is where the
concreteness, that I have referred to before, lies hidden; for in
situations where the truth or untruth of a proposition that involves
possible worlds (not just the truth of arithmetic statements) there
is a requirement of a concrete realization between the observer of
that world and the possible world. This latter idea is explicit in
the Everett and Rovelli Interpretations. (Concreteness is the
property of being “this and not anything else” that is invariant to
point of view.)
Perhaps. I would say that the concreteness is in both B and p (in Bp &
p). Also the duality above, the cartesian one, extends itself into the
duality between Bp & Dt and Bp & p & Dt (or Dp, it is equivalent).
The concreteness is then made even more concrete by imposing the
existence of a model satisfying the proposition (assuming the machine
talk first-order logic, Dt is equivalent with the existence of a model
by Gödel's completeness theorem).
But remember that, unlike Everett and Rovelli, we do not assume QM,
nor the existence of a primitive physical world. A concrete
realization is just a local and relative very probable (with the comp-
measure) universal number.
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