On 11 Jan 2012, at 19:35, acw wrote:

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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 an equivalence class. It seems to me that the concept of plurality of 1prequires a 3p to be coherent, but how does a 3p exist unless it isa 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 globally coherent(it is most certainly locally coherent), and may or may not becomputable, typically I imagine it as being locally computed by aninfinity of TMs, from the 1p. At least one coherent 3p foundationexists as the UD, but that's something very different from theuniverse a structural realist would believe in (for example, 'thisuniverse', or the MWI multiverse). So a coherent 3p foundationalways exists, possibly an infinity of them. The parts (or even thewhole) of the 3p foundation should be found 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's theoryis that an OM(Observer Moment) corresponds to a Sigma-1 sentence.

`You can ascribe a sort of local consciousness to the person living,`

`relatively to you, that Sigma_1 truth, but the person itself is really`

`related to all the proofs (in Platonia) of that sentences (roughly`

`speaking).`

Bruno

I 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) is correct, mathematics is the same for any system or being you can imagine.I am not sure why. "Sigma_1 arithmetic" would be the same; buthighermathematics (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 be NOTTuringemulable. Hypercomputation and infinite (human) minds wouldcontradictcomp, not CT. On the contrary, they need CT to claim that theycomputemore than any programmable machines. CT is part of comp, but comp is 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: all effectively computable functions are Turing-computable.I understand, but that is confusing. David Deutsch and manyphysicistsare a bit responsible of that confusion, by attempting to have anotionof "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 by theunique 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't reallyneedmore 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 for this(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 the ability to find himself in such a world (I'm leaning toward 'no, if it'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 something effectively computable that wasn't computable by a Turing machine.OK.In a way, that strong form of CT might already be false with comp,only in the 1p sense as you get a free random oracle as well asalwaysstaying 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 computer doesnotviolate CT can make us doubt about this.In the third person, there's no need to consider more than UD, which seems 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 in thecurrent CT would be present in hypercomputation (if one were toassumeit 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. Sotheconsistency 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) consequence ofCT,although Gödel's original proof does not use CT. provability, nor definability is not immune for diagonalization. Different machines proves 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 I maybewrong 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 even more theorems. 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 the truepropositions of arithmetic!!! Unfortunately it proves also allfalsepropositions of arithmetic. This follows easily by the secondtheorem ofGödel, because such a theory can prove its own consistency giventhat(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 it showssomenew truths, we lose the ability of telling true from false, makingita 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)? OrdidI 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 close to Cantor's paradise are accepted in the ontology, wouldn't that risk potentially 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 of entities appearing in the theory and that the possibility of concrete infinities appearing in the physics and thus leading to possible worlds where brains implementations have with concrete infinities = lack of substitution level.Remember that, like consistency, (~ probable comp) is consistentwithcomp. If comp is true, like consistency, it is not provable, andso 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 solid evidencesforCOMP being true(in the sense of no evidence about it being falsebeingfound, its predictions confirmed by observation and so on), wouldn't that warrant a belief in it? It would be religious belief in a way, like belief in the consistency of PA or some others belief in God or primitive matter or reality or particular physical laws or whatever (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 you cannotprove,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 PA consistency(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'm done reading 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) is computable.Here there is a difficulty, and many people get it wrong. Thereis afrequent error in logic which mirrors very well Searles error inhischinese 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 still nothave aclue about what the meaning of Einstein's answer. In fact I wouldjustmake 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 through itsprovability 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 PA canstillnot prove the consistency of PA.I don't think I said that PA can take other system's truths as theirown, even if it can simulate other systems, it cannot believe theyaretrue, and if it could, it no longer would be PA, but some othersystem.What I was talking about was a bit different, formal systems can besimulated by some UTM, thus while the UTM wouldn't "be" the system(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-emulable system, the answers will always be the same. That of course doesn'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 is not always 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_1 sentences(withthe shape ExP(x) with P decidable) she is universal, with respectto*computability. You can add as many axioms you want, the machinewillnot *compute* more functions. But adding axioms will always leadthemachine into proving more *theorems*.In AUDA, "belief" is modeled by provability (not computability),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 and ZF+(~C)proves exactly the same theorem in arithmetic, despite theyproves quitedifferent theorem about sets (so arithmetic is deeply independentof theaxiom of choice). ZF+k (= ZF + it exists an inaccessible cardinal) 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's provable)willdiffer per machine, and with adding more axioms, more beliefs are possible.There can be many 'believers'(axiomatic systems), but all of themcanbe 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 clearer versionof 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 when tryingtoconsider 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 in detail(assumingCOMP), a possible explanation for why it might not actually beuselessto perform such an experiment and why it might be a good practical test for verifying the consequences described in the UDA, various variations/factors/practicalities of that experiment are discussed(with goals such as reducing white rabbits, among a few others),somenot directly relevant to the argument and at the end I tried tosee 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 is assumedAND2) 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 clearer andshorterversion, with shorter sentences, without any digression, with clearsection 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 original ideaaspresented 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 any particular physical system, such as a CA. I then went on to say that if a 'physical CA' is chosen, there are many practical problems that could 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 something basedon aPA machine), which implements a message-passing operating system/scheduler on top of it. I argue that the careful use of a random oracle should reduce the chance of white-rabbits and increasethe overall world's stability(1p experiences not being too jumpy),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 I decidedtoconsider more practical details as well, because if one day therewillbe a computationalist doctor to say yes to and one wants to perform such 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.

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