On 22 Oct 2011, at 16:46, Stephen P. King wrote:

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How is a "space" defined in strictly arithmetic terms?Why do you want to define it in arithmetic. With comp, arithmeticcan be used for the ontology, but the internal epistemology needsmuch more. Remember that the tiny effective sigma_1 arithmeticalready emulate all Löbian machines like PA, ZF, etc. Numbers canuse sets to understand themselves.Yes, that numbers are what the Stone duality based idea assumes,but numbers alone do not induce the "understanding" unless anduntil sets are defined in distinction to them.Numbers alone are not enough, you need to make explicit theassumption of addition and multiplication. And that is provablyquite enough.To understand this you need to understand that the partialrecursive functions are representable in very tiny theories ofarithmetic (like Robinson Arithmetic), and you need to recall thatyou assume that you would survive with a digital brain/body. Therest 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 anotion of understanding obtain. Lobian machines are not just purearithmetic, 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 propositionsthan 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 mutuallynecessary axioms which can be forlaised with the use of very few(different) axioms. It is less ambiguous to measure the "force" ofa theory by the amount of arithmetic theorem they are able toprove. Note that ZF and ZFC (ZF + axiom of choice) have the exactsame force. The axiom of choice has no bearing on the arithmeticalreality. Of course, some proofs of arithmetical theorem can beshorter, but all proof using the axiom of choice can be donewithout 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 isexplicitly non-relativistic....I suggest you read the original long text by Everett, in the DeWittand Graham book. The fact that Everett shows this without assuminganything about relativity makes the case even stronger. But I don'tthink this is relevant on the topic. The digital mechanisthypothesis is neutral on physics. And the conclusion is that thewhole of physics is a number "illusion".Yes, and that is its failing. It takes the physical world to beepiphenomena.

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 Lobsense, 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" itsnon-standardness and thus it would bet that only it isrecursive, thus it's Bp&p would be 1p and not 3p truth.? (Bp&p) is the definition of 1-p in, well not really inarithmetic, but in terms of arithmetic. We cannot define "p" (pis 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 thatsuch undefinability is what makes truth behaving like a machine'sgod (like in Plato) and the knower like an inner God (like inPlato, or like with Plotinus' universal soul).Bruno http://iridia.ulb.ac.be/~marchal/I wish we could dispense with the idea of entities what areimpossible to exist.

I wish that too.

An entity cannot both be *all-knowing* and have an existence apartfrom the rest of the Totality.

It might be the price of being all knowing.

Theologies seriously need to be sure that their entities are notself-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/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.