Hi Stephen, On 13 Jan 2012, at 00:58, Stephen P. King wrote:

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Hi Bruno, On 1/12/2012 1:01 PM, Bruno Marchal wrote:On 11 Jan 2012, at 19:35, acw wrote:On 1/11/2012 19:22, Stephen P. King wrote:Hi,I have a question. Does not the Tennenbaum Theorem prevent theconceptof first person plural from having a coherent meaning, since itseems tomakes PA unique and singular? In other words, how can multiplecopies ofPA generate a plurality of first person since they would be anequivalence class. It seems to me that the concept of pluralityof 1prequires a 3p to be coherent, but how does a 3p exist unless itis a 1pin the PA sense? Onward! StephenMy understanding of 1p plural is merely many 1p's sharing anapparent 3p world. That 3p world may or may not be globallycoherent (it is most certainly locally coherent), and may or maynot be computable, typically I imagine it as being locallycomputed by an infinity of TMs, from the 1p. At least one coherent3p foundation exists as the UD, but that's something verydifferent from the universe a structural realist would believe in(for example, 'this universe', or the MWI multiverse). So acoherent 3p foundation always exists, possibly an infinity ofthem. The parts (or even the whole) of the 3p foundation should befound within the UD.As for PA's consciousness, I don't know, maybe Bruno can say a lotmore about this. My understanding of consciousness in Bruno'stheory is that an OM(Observer Moment) corresponds to a Sigma-1sentence.You can ascribe a sort of local consciousness to the person living,relatively to you, that Sigma_1 truth, but the person itself isreally related to all the proofs (in Platonia) of that sentences(roughly speaking).OK, but that requires that I have a justification for a belief inPlatonia. The closest that I can get to Platonia is something likethe class of all verified proofs (which supervenes on some form ofphysical process.)

`You need just to believe that in the standard model of PA a sentence`

`is true or false. I have not yet seen any book in math mentioning`

`anything physical to define what that means.`

*All* math papers you cited assume no less.

I simply cannot see how Sigma_1 sentences can interface with eachother such that one can "know" anything about another absent someform of physicality.

`The "interfaces" and the relative implementations are defined using`

`addition and multiplication only, like in Gödel's original paper. Then`

`UDA shows why physicality is an emergent pattern in the mind of`

`number, and why it has to be like that if comp is true. AUDA shows how`

`to make the derivation.`

If I take away all forms of physical means of communicating ideas,no chalkboards, paper, computer screens, etc., how can ideas bepossibly communicated?

`Because arithmetical truth contains all machine 'dreams", including`

`dreams of chalkboards, papers, screens, etc. UDA has shown that a`

`"real paper", or & "real screen" is an emergent stable pattern`

`supervening on infinities of computation, through a competition`

`between all universal numbers occurring below our substitution level.`

`You might try to tell me where in the proof you lost the arguement.`

Mere existence does not specify properties.

`That's not correct. We can explain the property "being prime" from the`

`mere existence of 0, s(0), s(s(0)), ... and the recursive laws of`

`addition and multiplication.`

I go so far as considering that the wavefunction and its unitaryevolution exists and it is a sufficiently universal "physical"process to implement the UD, but the UD as just the equivalent toIntegers, nay, that I cannot believe in. “One cannot speak aboutwhatever one cannot talk.” ~ Maturana (1978, p. 49)

`I think Maturana was alluding to Wittgenstein, and that sentence is`

`almost as ridiculous as Damascius saying "one sentence about the`

`ineffable is one sentence too much". But it is a deep meta-truth`

`playing some role in number's theology.`

`But I think that you cannot define the universal wave without`

`postulating arithmetical realism. In fact real number+trigonometrical`

`function is a stronger form of realism than arithmetical realism.`

`Adding "physical" in front of it adds nothing but a magical notion of`

`primary substance. Epistemologically it is a form of treachery, by`

`UDA, it singles out a universal number and postulate it is real, when`

`comp explains precisely that such a move cannot work.`

Bruno

BrunoI think you might be confusing structures/relations which can becontained within PA with PA itself.On 1/11/2012 12:07 PM, acw wrote:On 1/10/2012 17:48, Bruno Marchal wrote:On 10 Jan 2012, at 12:58, acw wrote:On 1/10/2012 12:03, Bruno Marchal wrote:On 09 Jan 2012, at 19:36, acw wrote:To put it more simply: if Church Turing Thesis(CTT) iscorrect,mathematics is the same for any system or being you canimagine.I am not sure why. "Sigma_1 arithmetic" would be the same;but highermathematics (set theory, analysis) might still be different.If it's wrong, maybe stuff like concrete infinities,hypercomputationand infinite minds could exist and that would falsify COMP,howeverthere is zero evidence for any of that being possible.Not sure, if CT is wrong, there would be finite machines,working infinite time, with well defined instructions, which would beNOT Turingemulable. Hypercomputation and infinite (human) minds wouldcontradictcomp, not CT. On the contrary, they need CT to claim thatthey computemore than any programmable machines. CT is part of comp, butcompis not part of CT. Beyond this, I agree with your reply to Craig.In that response I was using CT in the more unrestricted form:alleffectively computable functions are Turing-computable.I understand, but that is confusing. David Deutsch and manyphysicistsare a bit responsible of that confusion, by attempting to havea notionof "effectivity" relying on physics. The original statement ofChurch,Turing, Markov, Post, ... concerns only the intuitively humancomputablefunctions, or the functions computable by finitary means. Itassertsthat the class of such intuitively computable functions is thesame asthe class of functions computable by some Turing machine (or bytheunique universal Turing machine). Such a notion is a prioricompletelyindependent of the notion of computable by physical means.Yes, with the usual notion of Turing-computable, you don'treally needmore than arithmetic.It might be a bit stronger than the usual equivalency proofsbetween avery wide range of models of computation (Turing machines,Abacus/PAmachines, (primitive) recursive functions (+minimization), allkindsof more "current" models of computation, languages and so on).Yes. I even suspect that CT makes the class of functionscomputable byphysics greater than the class of Church.That could be possible, but more evidence is needed forthis(beyondthe random oracle). I also wonder 2 other things: 1) would we beableto really know if we find ourselves in such a world (I'm leaning toward unlikely, but I'm agnostic about this) 2) would someone performing my experiment(described in another message), lose theability to find himself in such a world (I'm leaning toward 'no,ifit's possible now, it should still be possible').If hypercomputation was actually possible that would mean thatstrongvariant of CT would be false, because there would be somethingeffectively computable that wasn't computable by a Turingmachine.OK.In a way, that strong form of CT might already be false withcomp,only in the 1p sense as you get a free random oracle as wellas alwaysstaying consistent(and 'alive'), but it's not false in the 3pview...Yes. Comp makes physics a first person plural reality, and apriori wemight be able to exploit the first plural indeterminacy tocompute morefunction, like we know already that we have more "processes",like thatfree random oracle. The empirical fact that quantum computerdoes notviolate CT can make us doubt about this.In the third person, there's no need to consider more than UD,whichseems to place some limits on what is possible, but in the first person, the possibilities are more plentiful (if COMP).Also, I do wonder if the same universality that is present inthecurrent CT would be present in hypercomputation (if one wereto assumeit would be possible)Yes, at least for many type of hypercomputation, notably of theform ofcomputability with some oracle.- would it even retain CT's current "immunity" fromdiagonalization?Yes. Actually the immunity of the class of computable functionsentailsthe immunity of the class of computable functions with oracle.So theconsistency of CT entails the consistency of some super-CT forlargerclass. But I doubt that there is a super-CT for the class offunctionscomputable by physical means. I am a bit agnostic on that.OK, although this doesn't seem trivial to me.As for the mathematical truth part, I mostly meant that from theperspective of a computable machine talking about axiomaticsystems -as it is computable, the same machine (theorem prover) wouldalwaysyield the same results in all possible worlds(or shared dreams).I see here why you have some problem with AUDA (and logic). CT= thenotion of computability is absolute. But provability is notabsolute atall. Even with CT, different machine talking or using different axiomatic system will obtain different theorems.In fact this is even an easy (one diagonalization) consequenceof CT,although Gödel's original proof does not use CT. provability, nordefinability is not immune for diagonalization. Differentmachinesproves different theorems.If questions(axiomatic system being looked into) are different, results can be different too, but if the questions (and inference steps) are the same, shouldn't the result always be the same? This would follow from CTT implementing a theorem prover.Although with my incomplete understanding of the AUDA, and Imay bewrong about this, it appeared to me that it might be possiblefor amachine to get more and more of the truth given the consistency constraint.That's right both PA + con(PA) and PA + ~con(PA) proves more true arithmetical theorems than PA.And PA + con(PA + con(PA + con (PA + con PA)) will proves evenmoretheorems. The same with the negation of those consistency.I do wonder what kind of interesting theorems would result fromthose.Note that the theory PA* = PA* + con(PA*), which can be definedfinitelyby the use of the Kleene recursion fixed point proves ALL thetruepropositions of arithmetic!!! Unfortunately it proves also allfalsepropositions of arithmetic. This follows easily by the secondtheorem ofGödel, because such a theory can prove its own consistencygiven that(con "itself") is an axiom, and by Gödel II, it is inconsistent.PA* is the one which claims its on consistency as a theorem? Itwouldbe inconsistent in that case, and unfortunately, even if itshows somenew truths, we lose the ability of telling true from false,making ita bit useless once one starts making inferences from falsestatements.But, yes, once you have a consistent machine, you can extend its provability ability on the whole constructive transfinite.How come? PA being able to encode a theorem prover for ZFC,wouldn'tmean PA believes in the truth of ZFC(or some other set theory)?Or didI misunderstand something here. Maybe you should recommend me some book or paper to read that would explain this to me.As for higher math, such as set theories: do they have a modeland arethey consistent? (that's an open question) If some forms closetoCantor's paradise are accepted in the ontology, wouldn't thatriskpotentially falsifying COMP?Trivially. You can refute comp in the theory ZF + ~comp(accepting someformalization of comp in set theory).With that question I was more concerned about the number ofentitiesappearing in the theory and that the possibility of concrete infinities appearing in the physics and thus leading to possibleworlds where brains implementations have with concreteinfinities =lack of substitution level.Remember that, like consistency, (~ probable comp) isconsistent withcomp. If comp is true, like consistency, it is not provable,and so youcan add (~ provable comp), or (con ~comp) to the comp theory ofeverything (arithmetic) without getting an inconsistency withcomp. Ofcourse you should not add ~comp to comp at the same(meta)level. Youwill get a contradiction.I've always wondered about one thing: if there are solidevidences forCOMP being true(in the sense of no evidence about it being falsebeingfound, its predictions confirmed by observation and so on),wouldn'tthat warrant a belief in it? It would be religious belief in away,like belief in the consistency of PA or some others belief inGod orprimitive matter or reality or particular physical laws orwhatever(some may be true, other may be false, but one must have some assumptions if they have to bet on something). In another way, a conscious machine can always doubt that it's a machine and this doubt would not make it inconsistent.Likewise, as above, the theory PA + (PA is inconsistent) isconsistent.You can prove, in it, that the false is provable, but youcannot prove,in it, the false. Bf -> f is not a theorem (Bf -> f) = ~Bf = consistency.I always found that as a non-intuitive consequence of Godel'stheorems.PA doesn't contains a proof of its own consistency, if it'sconsistent(it is inconsistent if it does contain it).On the other hand, in a stronger theory, which shows PAconsistency(for example ZFC), wouldn't PA + (PA is inconsistent) beinconsistent?Or to put it different, wouldn't it lack a model (because 'PA is inconsistent' would be false, despite not being provable within PA itself)?Maybe it could even be generalized to adding a sentence whichcould befalse, but from which nothing false could be proven within some axiomatic system.I hope that some of these ideas will be clearer to me after I'mdonereading those books on logic and provability.I can see many reasons why a particular machine/system wouldwant totalk about such higher math, but I'm not sure how it could endup withdifferent discourses/truths if the machine('s body) iscomputable.Here there is a difficulty, and many people get it wrong. Thereis afrequent error in logic which mirrors very well Searles errorin hischinese room argument. With comp I can certainly simulateEinstein'sbrain, but that fact does not transform me into Einstein. Ifsomeoneasks me a complex question about relativity, I might be able toanswerby simulating what Einstein would respond, but I might stillnot have aclue about what the meaning of Einstein's answer. In fact Iwould justmake it possible for Einstein to answer the question. Not me. Like wise, a quasi debilitating arithmetical theorem system likeRobinson Arithmetic, which cannot prove x+y = y+x, for example,is stillTuring universal, and as such can imitate PA perfectly throughitsprovability abilities. That is, RA is able to prove that PA canprovex+y=y+x. But RA has not the power to be convinced by that PA'sproof,like I can simulate Einstein without having the gift tounderstand anywords by him. RA can simulate PA and ZF, and even ZF+k (which can prove theconsistency of ZF), but this does not give to RA the*provability* powerof PA, ZF or ZF+k.PA can prove that ZF can prove the consistency of PA, but PAcan stillnot prove the consistency of PA.I don't think I said that PA can take other system's truths astheirown, even if it can simulate other systems, it cannot believethey aretrue, and if it could, it no longer would be PA, but some othersystem.What I was talking about was a bit different, formal systems canbesimulated by some UTM, thus while the UTM wouldn't "be" thesystem (inthe same way that in COMP, a brain could allow a mind to manifest relatively to some other observers, but it wouldn't be that mind), given the same questions asked to some particular Turing-emulablesystem, the answers will always be the same. That of coursedoesn'tmean that PA can take ZFC truths as its own - they are notprovable inPA.In another way, the totality of possible discourses should bepresentwithin PA, but that doesn't mean PA can take them as its truths. There's another problem here: not all formal systems will be consistent or have models, but unless we have proofs of their inconsistency, we will never know.Some of this confusion might be because, we as humans, sometimestakeother people or system's beliefs as our own, even if doing so isnotalways rational, but despite that this increases the risk of being wrong, it also lets us get to a lot more truth.I can see it discovering the independence of certain axioms (forexample the axiom of choice or the continuum hypothesis), butwouldn'tall the math that it can /talk/ about be the same? The machinewouldhave to assume some axioms and reason from there.Yes. And with different axioms you get different provabilityaptitudes.Once a machine can prove all true arithmetical sigma_1sentences (withthe shape ExP(x) with P decidable) she is universal, withrespect to*computability. You can add as many axioms you want, themachine willnot *compute* more functions. But adding axioms will alwayslead themachine into proving more *theorems*.In AUDA, "belief" is modeled by provability (notcomputability), andthen knowledge is defined in the usual classical (Theaetetus)ways. Allbeliefs of the correct machines will obeys to the same self-referentiallogic, but all belief-content will differ from a machine to adifferentmachine. PA and ZF have the same self-referential logics, butthey havequite different belief, even restricted on the numbers.For example ZF proves more arithmetical truth than PA. ZFC andZF+(~C)proves exactly the same theorem in arithmetic, despite theyproves quitedifferent theorem about sets (so arithmetic is deeplyindependent of theaxiom of choice). ZF+k (= ZF + it exists an inaccessiblecardinal)proves *much more* arithmetical theorem than ZF. To sum up: Computability is an absolute notion. Provability is a relative notion.I think we're mostly in agreement here, beliefs (what'sprovable) willdiffer per machine, and with adding more axioms, more beliefs are possible.There can be many 'believers'(axiomatic systems), but all ofthem canbe implemented by the same base(their "body", some theorem prover implemented by some UTM).However, I would like to know what the many more arithmeticaltheoremsZF(and some of its extensions) that proves are. The only ones I'mfamiliar with are the type of PA's consistency and Goodstein'stheoremas well as similar results about the fact that some sequences terminate/halt.BTW, acw, you might try to write a shorter and clearerversion of yourjoining post argument. It is hard to follow. If not, I mighttake muchmore time. BrunoI think I talked about too many different things in that post,not alldirectly relevant to the argument (although relevant whentrying toconsider as many consequences as possible of that experiment).If someparts are unclear, feel free to ask in that thread. The generaloutline of what I talked about in the part you have yet tocomment on:a generalized form of the experiment the main character from"Permutation City" novel performed is described indetail(assumingCOMP), a possible explanation for why it might not actually beuselessto perform such an experiment and why it might be a goodpracticaltest for verifying the consequences described in the UDA,variousvariations/factors/practicalities of that experiment arediscussed(with goals such as reducing white rabbits, among a fewothers), somenot directly relevant to the argument and at the end I triedto see ifthe notion of observer can be better defined and tried to showthatthe notion of "generalized brain" might not always be anappropriateway to talk about an observer. That post was mostly meant to beexploratory and I hoped the ensuing discussion would lead to 2things:1) assessing the viability of that experiment if COMP isassumed AND2) reaching a better definition of the notion of observer.OK. This I think I understood this, but your style is not easy,and itmight be useful, even within your goal, to work on a clearerand shorterversion, with shorter sentences, without any digression, withclearsection and subsection, so as to invite most people (includingme) tograsp it, or to find a flaw, and this in reasonable time. InparticularI fail to see the point of discussing the use of differentuniversalsystems like you did with the Cellular Automata (CA). Bruno http://iridia.ulb.ac.be/~marchal/I will consider rewriting it if the time allows. The originalidea aspresented in that book, had a CA as their "primary physics", so Itried to show the difference between my experiment and theexperimentin the book. If one assumes COMP, there is no longer a need for choosing anyparticular physical system, such as a CA. I then went on to saythatif a 'physical CA' is chosen, there are many practical problemsthatcould occur, both a decrease in stability(white rabbits,jumpyness),as well as many problems for those living with the system(speed-of-light limit and most social problems present in ourphysicalworld). I think it could only be 100% stable if the (mind's) substitutionlevel is exactly at the CA level, which was not the case in thebook.Instead of such a "primary physics", I just choose an easy todesign/program Turing-equivalent machine (such as somethingbased on aPA machine), which implements a message-passing operating system/scheduler on top of it. I argue that the careful use of arandom oracle should reduce the chance of white-rabbits andincreasethe overall world's stability(1p experiences not being toojumpy), butI still don't know to what degree this would be sufficient (oracle implemented by dovetailing or leaving "undefined"). I tried to consider how the choice of OS and observer's implementation could affect the world's stability(1p).As a thought experiment, I could have stopped there, but Idecided toconsider more practical details as well, because if one daythere willbe a computationalist doctor to say yes to and one wants toperformsuch an experiment, they better have all their details right,otherwise after performing the experiment, their future 1pexperiencesmight not be very pleasant.--You received this message because you are subscribed to the GoogleGroups "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.http://iridia.ulb.ac.be/~marchal/--You received this message because you are subscribed to the GoogleGroups "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.

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