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).



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.



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