On 10/22/2011 8:23 AM, Bruno Marchal wrote:

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 you proposed is more 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 generated andexecuted. To prefix them is usefulm when they are generated by arandom 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") are surely still 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 "computationalneighborhoods" are a priori uncoutable. That fits with thetopological 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 themechanist 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 ormachine perceptions, themselves belonging to the category of numberrelation. To *assume* space and time in the ontology can only bemisleading, with respect to the mechanist formulation of the mind-bodyproblem.

[SPK]

`I am thinking of mathematical spaces, not the physical space of`

`experience.`

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 assumptionof addition and multiplication. And that is provably quite enough.To understand this you need to understand that the partial recursivefunctions are representable in very tiny theories of arithmetic (likeRobinson Arithmetic), and you need to recall that you assume that youwould survive with a digital brain/body. The rest follows from the UDreasoning.

That is not addressing my point.

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.`

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 are verynice, they are machine/theory independent, and that is quite inthe spirit of comp, but it seems to be usable only in ASSA typeapproach. I do not exclude this can help for providing a role tolittle program, but I don't see at all how it could help for thecomputation of the first person indeterminacy, aka the derivationof physics from computer science needed when we assume comp incognitive science. In the work using Solomonof-Levin, themind-body problem is still under the rug. They don't seem aware ofthe 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 them being generated in a UD, that structure will be lost in the process of observation. The net effect is that UD* is just as much a "veil" on the 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 is theeverything for the computational approach, but it is tiny comparedto the first person view of it by internal observers accounted inthe 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, giventhat 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" of atheory 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. Ofcourse, some proofs of arithmetical theorem can be shorter, but allproof using the axiom of choice can be done without using the axiomchoice.

I wish I could find a broader discussion of that claim.

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 will alwaysseethe Solomonoff-Levin distribution, or universal prior.I don't think it makes sense to use a universal prior. That would make sense if we suppose there are computable universes, and if we try to measure the probability we are in such structure. This is typical of Schmidhuber's approach, which is still quite similar to physicalism, where we conceive observers as belonging to computableuniverses. 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 is offarmore general applicability than Schmidhuber's model. There neednot beany identity thesis invoked, as for example in applications such as observers of Rorshach diagrams. And as for identity thesis, you do have a type of identity thesis in the 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 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 mechanist hypothesisis neutral on physics. And the conclusion is that the whole of physicsis a number "illusion".

`Yes, and that is its failing. It takes the physical world to be`

`epiphenomena. Why does the physical world even need to exist at all?`

There has to be some form of identity thesis between brain and mindthat 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 entirely bythe (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 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?`

The proof would require showing that a Lobian machine on anon-standard model of arithmetic would *not* be able to "see" itsnon-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 anyeffective 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's god(like in Plato) and the knower like an inner God (like in Plato, orlike 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. An entity cannot both be *all-knowing* and have an`

`existence apart from the rest of the Totality. Theologies seriously need`

`to be sure that their entities are not self-contradictory.`

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