On 22 Oct 2011, at 16:46, Stephen P. King wrote:
How is a "space" defined in strictly arithmetic terms?
Why do you want to define it in arithmetic. With comp, arithmetic
can be used for the ontology, but the internal epistemology needs
much more. Remember that the tiny effective sigma_1 arithmetic
already emulate all Löbian machines like PA, ZF, etc. Numbers can
use sets to understand themselves.
Yes, that numbers are what the Stone duality based idea assumes,
but numbers alone do not induce the "understanding" unless and
until sets are defined in distinction to them.
Numbers alone are not enough, you need to make explicit the
assumption of addition and multiplication. And that is provably
quite enough.
To understand this you need to understand that the partial
recursive functions are representable in very tiny theories of
arithmetic (like Robinson Arithmetic), and you need to recall that
you assume that you would survive with a digital brain/body. The
rest follows from the UD reasoning.
That is not addressing my point.
What is your point?
My point is that we don't need sets to have dreaming machines (notably
dreaming about sets, and perhaps making good use of them).
This is why it cannot be monistic above the "nothing" level.
?
Tarski's theorem prevents understanding in number monist theories.
?
If arithmetic truth cannot be defined in arithmetic, how can a
notion of understanding obtain. Lobian machines are not just pure
arithmetic, it seems.
It is pure arithmetic. It is the very idea of Gödel's arithmetization
of metamathematic. The study of Löbian machines can be seen as the
study of what numbers can prove about themselves and their points of
view by using nothing more than addition and multiplication + a bit of
classical logic (which is part of the number, by the arithmetization).
How could a Löbian machine grasp "arithmetical truth"? Well, she
can't. But she can define truth for all Sigma-i or Pi_i truth notions,
and she can approximate the whole truth, talk indirectly about it, or
intuit stronger arithmetical or mathematical axioms, and transform
herself.
Tiny as opposed to ???
To big! Or strong. ZF proves much more arithmetical propositions
than PA.
Oh, the number of independent but mutually necessary axioms?
This would not been a good measure of complexity or strongness,
given that you can find theories with many independent and mutually
necessary axioms which can be forlaised with the use of very few
(different) axioms. It is less ambiguous to measure the "force" of
a theory by the amount of arithmetic theorem they are able to
prove. Note that ZF and ZFC (ZF + axiom of choice) have the exact
same force. The axiom of choice has no bearing on the arithmetical
reality. Of course, some proofs of arithmetical theorem can be
shorter, but all proof using the axiom of choice can be done
without using the axiom choice.
I wish I could find a broader discussion of that claim.
I proved this as an exercise in set theory when student. Hint: use
Gödel constructible sets (cf V = L). Buy the very good book by Krivine
on Set Theory. But this is something technical about two particular
LUMs, and has no bearing with the topic, I think. ZF + k (ZF + the
existence of an accessible cardinal) proves much more arithmetical
propositions than ZF and ZFC, for example the arithmetization of ZF
consistency).
How so? Please point to a discussion of this! Everett is
explicitly non-relativistic....
I suggest you read the original long text by Everett, in the DeWitt
and Graham book. The fact that Everett shows this without assuming
anything about relativity makes the case even stronger. But I don't
think this is relevant on the topic. The digital mechanist
hypothesis is neutral on physics. And the conclusion is that the
whole of physics is a number "illusion".
Yes, and that is its failing. It takes the physical world to be
epiphenomena.
Why epiphenomena? Why not phenomena?
Why does the physical world even need to exist at all?
Its phenomenological existence is a theorem in the number's theory of
number's dreams.
The idea is that every 1p would observe itself, in the Lob
sense, to be recursive.
?
How does a Lobian machine recognize its properties?
Which properties. I'm sorry but you are losing me.
Any of its properties. How does a Lobian machine know what it is,
even if incompletely?
By the numbers (machines, programs, ...) self-referential abilities.
You might read the paper by Smorynski: 50 years of arithmetical self-
references. See the general biblio of Conscience et Mécanisme.
The proof would require showing that a Lobian machine on a non-
standard model of arithmetic would *not* be able to "see" its
non-standardness and thus it would bet that only it is
recursive, thus it's Bp&p would be 1p and not 3p truth.
? (Bp&p) is the definition of 1-p in, well not really in
arithmetic, but in terms of arithmetic. We cannot define "p" (p
is true) in any effective theory.
This is part of the incompleteness of your result :-(
It is part of Tarski undefinability result, and it concerns *all*
effective theories, and *all* machines interested in searching truth.
And actually, it is a big chance for machine's theology, given that
such undefinability is what makes truth behaving like a machine's
god (like in Plato) and the knower like an inner God (like in
Plato, or like with Plotinus' universal soul).
Bruno
http://iridia.ulb.ac.be/~marchal/
I wish we could dispense with the idea of entities what are
impossible to exist.
I wish that too.
An entity cannot both be *all-knowing* and have an existence apart
from the rest of the Totality.
It might be the price of being all knowing.
Theologies seriously need to be sure that their entities are not
self-contradictory.
Sure. But that's what the arithmetical interpretation of Plotinus
offers at the least. A proof of of the consistency of Plotinus'
theology with respect to arithmetic and digital mechanism. It offers
also a physic on a plate, so we can test comp, and this interpretation
of Plotinus.
Bruno
http://iridia.ulb.ac.be/~marchal/
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