On 21 Oct 2011, at 20:34, Stephen P. King wrote:

On 10/21/2011 10:10 AM, Bruno Marchal wrote:On 21 Oct 2011, at 15:08, Stephen P. King wrote:On 10/21/2011 8:14 AM, Bruno Marchal wrote:On 19 Oct 2011, at 05:30, Russell Standish wrote:On Mon, Oct 17, 2011 at 07:03:38PM +0200, Bruno Marchal wrote:This, ISTM, is a completely different, and more wonderfulbeast, thanthe UD described in your Brussells thesis, or Schmidhuber's '97 paper. This latter beast must truly give rise to a continuum of histories, due to the random oracles you were talking about.All UDs do that. It is always the same beast.On reflection, yes you're correct. The new algorithm youproposed ismore efficient than the previous one described in your thesis, asmachines are only executed once for each prefix, rather over andoveragain for each input having the same prefix. But in anenvironment ofunbounded resources, such as we're considering here, that has noimport.Note that my programs are not prefixed. They are all generatedand executed. To prefix them is usefulm when they are generatedby a random coin, which I do not need to do.So the histories, we're agreed, are uncountable in number, but OMs(bundles of histories compatible with the "here and now") aresurelystill countable.This is not obvious for me. For any to computational states whichare in a sequel when emulated by some universal UM,there areinfinitely many UMs, including one dovetailing on the reals,leading to intermediate states. So I think that the"computational neighborhoods" are a priori uncoutable. That fitswith the topological semantics of the first person logics (S4Grz,S4Grz1, X, X*, X1, X1*). But many math problems are unsolved there.Hi Bruno and Russel,I would like to better understand what "topological semantics"means. Are you considering relations defined only in settheoretical sense, ala the closed or open or clopen nature of thesets relative to each other?I guess you know the topological semantics of intuitionist logic.Instead of interpreting the propositions by sets in a booleanalgebra, you interpret it by open set in a topological space. Youhave soundness and completeness theorem for that. I think this camefrom semantics for the semantics for the modal logic S4, whichmirrors well intuitionist logic.What about the form of the axiom of choice for the set theory??There are multiple versions of set theory depending on theselection of its form of axiom of choice.

`There are many set theories at the start, and they are not equivalent.`

`Set theories assumed too much, especially with respect to the`

`mechanist assumptions.`

How do you induce compactness?Why do you want the space being compact. The point of thetopological semantics is that it works for all topological spaces,like any boolean algebra can be used for classical logic.A space needs to be compact for many reasons including, but notexclusive to, the ability to have a physics in them. Spaces that arenot compact will not have, for instance, fixed points that allow fornotions such as centers of mass, etc. There are also analycityreasons, and more.

`UDA shows that space and time belongs to the category of mind or`

`machine perceptions, themselves belonging to the category of number`

`relation. To *assume* space and time in the ontology can only be`

`misleading, with respect to the mechanist formulation of the mind-body`

`problem.`

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 LĂ¶bian 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 and untilsets 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.`

This is why it cannot be monistic above the "nothing" level.

?

Tarski's theorem prevents understanding in number monist theories.

?

If we take the no information ensemble,You might recall what you mean by this exactly.and transform it by applying auniversal turing machine and collect just the countable outputstringwhere the machine halts, then apply another observer function that also happens to be a UTM, the final result will still be a Solomonoff-Levin distribution over the OMs.This is a bit unclear to me. Solomonof-Levin distribution arevery nice, they are machine/theory independent, and that is quitein the spirit of comp, but it seems to be usable only in ASSAtype approach. I do not exclude this can help for providing arole to little program, but I don't see at all how it could helpfor the computation of the first person indeterminacy, aka thederivation of physics from computer science needed when we assumecomp in cognitive science. In the work using Solomonof-Levin, themind-body problem is still under the rug. They don't seem awareof the first/third person description.S-L seems to assume 1p = 3p or no 1p at all!I think they just ignore 1p.This result follows fromthe compiler theorem - composition of a UTM with another one isstilla UTM.So even if there is a rich structure to the OMs caused by thembeinggenerated in a UD, that structure will be lost in the process ofobservation. The net effect is that UD* is just as much a "veil"onthe ultimate ontology as is the no information ensemble.UD*, or sigma_1 arithmetic, can be seen as an effective(mechanically defined) definition of a zero information. It isthe everything for the computational approach, but it is tinycompared to the first person view of it by internal observersaccounted in the limit by the UD.How do we define this notion of size?There are many ways. Tiny in the sense of the ordering of sets ofprovable proposition of arithmetic, for example. Robinson theory istiny compared to PA which is tiny compared to ZF, in that sense.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 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.`

Unless I'm missing something here.Lets leave the discussion of the universal prior to anotherpost. In anutshell, though, no matter what prior distribution you put onthe "noinformation" ensemble, an observer of that ensemble willalways seethe Solomonoff-Levin distribution, or universal prior.I don't think it makes sense to use a universal prior. That wouldmake sense if we suppose there are computable universes, and ifwetry to measure the probability we are in such structure. This istypical of Schmidhuber's approach, which is still quite similartophysicalism, where we conceive observers as belonging tocomputableuniverses. Put in another way, this is typical of using somesort ofidentity thesis between a mind and a program.I understand your point, but the concept of universal prior isof farmore general applicability than Schmidhuber's model. There neednot beany identity thesis invoked, as for example in applications suchasobservers of Rorshach diagrams.And as for identity thesis, you do have a type of identitythesis inthe statement that "brains make interaction with other observers relatively more likely" (or something like that).yes, by the duplication (multiplication) of populations ofobservers, like in comp, but also like in Everett.But we also need something that acts to code the "no preferredreference frame" of GR. Everett does not solve this problem, itonly compounds it. :-(I am not sure I follow you. Everett makes clear that the choice ofbase is a false problem.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 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".`

There has to be some form of identity thesis between brain andmindthat prevents the Occam catastrophe, and also prevent the fullretreatinto solipsism. I think it very much an open problem what that is.This will depend on the degree of similarity between betweenquantum mechanics and the comp physics, which is given entirelyby the (quantified) material hypostases (mainly the Z1* and X1*logics). Open but well mathematically circumscribed problem.From what I have studied so far, the (static) relationshipbetween QM and COMP physics is the relationship between Booleanalgebras and Orthocomplete lattices.Let us hope. Even just that has not been proved. Open problem.I think that a partial solution might require a weakening of thedefinition of a countable model of arithmetic and application ofthe Tennenbaum theorem (that would allow for a variationalprinciple, something like q-deformed theories where q is a measureof the relative recursiveness of the model of the theory).?I am still studying the math on this, the problem that I have isthat there do not seem to be words for the idea that I see. Theclosest concept is how a fixed point is defined on a manifold...Crudely, think of a manifold (not a space per se) where every pointis a model of arithmetic, the fixed point is the one that iscountable and recursive.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.

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, thusit'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" (p istrue) 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/ -- 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.