On 28 Dec 2010, at 22:43, Brent Meeker wrote:

On 12/28/2010 1:13 PM, Bruno Marchal wrote:On 28 Dec 2010, at 19:46, Brian Tenneson wrote:Thank you, happy new year to you, too!Thanks.On Dec 27, 8:36 am, Bruno Marchal <marc...@ulb.ac.be> wrote:On 26 Dec 2010, at 22:51, Brian Tenneson wrote:"Limits To Science: God, Godel, Gravity"http://www.science20.com/hammock_physicist/limits_science_god_godel_g...Here is my comment: An important question is whether or not a TOE will be finite in length.Of course, this is a matter of definition.Indeed.I am taking 'TOE' to be, as a working definition, a completedescription of reality or a complete description of everythingthatexists.That does not exist. Arithmetical truth is already not recursively enumerable.If a complete description of arithmetical truth is not possible,whatexactly are we talking about?We talk about the set of all true arithmetical statements; writtenin the (first order logic) language of arithmetic.That set contains the verifiable atomic formula like the usualtaught in school: 0 + 1 = 1, 45 + 9 = 54, 3*7 + 3 = 24, etc. Where3 abbreviates s(s(s(0))), etc.Then the sigma_1 true sentence like Ex( x = s(0)), or ExEy(x = 2y).The existence of a computational state relatively to a universalnumber is sigma_1. Then the true pi_1 sentences like Ax(x = 0 Vnot(x = 0)).Here we can already show that for any (proving/believing) machinethere is an arithmetical pi_1 proposition which is true but notprovable by that machine. Note that the Riemann hypothesis can beshown to be equivalent with an arithmetical pi_1 sentence, and isan example of a pi_1 sentence that we suspect to be true but thatwe can still not prove.Then the sigma_2 sentences, equivalent to a proposition with theshape ExAyP(x, y, a, b, ...), then the pi_2 (AxEyP(x, y, a, b, ...)with P a decidable (sigma_0) propositions, etc. Most of them arebeyond the human mind.Arithmetical truth is the set of all true propositions inarithmetic. That set is not recursively enumerable (cannot bedefined by a sigma_1 proposition, and actually is not arithmetical(cannot be defined by *any* arithmetical proposition!). Thatfollows from Gödel and Tarski.We, humans, have a rather good intuition of what is a truearithmetical sentence, independently of the fact that we have torecognize that it can be quite tricky to decide if this or thatarithmetical proposition is true or not.Then one *might* consider this program which generates a TOE toarbitrary precision to be "the" TOE, a compression of aninfinitelylong document into a finitely long document, thus showing thatrealityat its core does not possess the trait of Kolmogorov randomness.Beingthat reality contains the uncomputable, it *seems* unlikely that everything can be finitely describable.This is simply wrong.That's good!The argument is made that a TOE can be in the form of a logicalstructure which is a tuple consisting of an underlying set, aset ofdistinguished constants (like zero), functions (like successor),andrelations (like less than) on this underlying set. Making the additional assumption that if there is a structure such that *all* logical structures can be "embedded" within it, then this type of universality endows such a structure with the same structure asreality. Thus this sort of ultimate structure would be in anintuitivesense like ultimate reality. Thus a description of this ultimate structure would be a description of reality.You have to distinguish the outside and inside view of the ultimate reality.What is the outside view of the ultimate reality?Assuming mechanism, arithmetical truth is enough. It is infiniteand non algorithmically generable.It is just absolutely undecidable that there is anything moreneeded for the outside view. So by Occam razor ...If we are "machine" (survive at some substitution for some level,or "are fully turing emulable" with "fully" meaning thatconsciousness is preserved) then the inside "knowledge" view ofarithmetical truth will be FAR BIGGER than arithmetical truth.Indeed, it corresponds plausibly to George Levy first personplenitude. That things is much less definable (if I can say) thanarithmetical truth. There is here an analog of Skolem paradox, withthe existence of a structure which looks small from outside and isimmensely big from inside.I don't understand how "a view of arithemetical truth" can be biggerthan arithmetical truth - unless the view includes some arithmeticaluntruths.

`The inside view might include some arithmetical untruth, but in that`

`case it can be shown that the machine will lose its Löbianity, so this`

`is not the interesting case. (think of PA + beweisbar('false')).`

`It is more related to the fact that the inside (first person) view`

`includes some non arithmetical truth. This is simpler to explain with`

`the definition of the first person view in AUDA. The first person is`

`the knower and it is defined by Bp & p, that is: the conjunction of`

`provability and truth. But truth is already non arithmetical, making`

`the knower already not definable by the first person, that is by`

`itself. In that sense the knower feels as being something different`

`and bigger from any definition it can handle, including "Bp". That is`

`actually why Lucas and Penrose infer from Gödel that we are not`

`machine, but this is incorrect, because if that was correct we would`

`conclude that machine could prove correctly that they are not machine.`

`Another way to illustrate this is by using Skolem paradox in set`

`theory. It is not a contradiction: just an amazing fact.`

`Consider the set theory ZF. It is a first order logical theory, so`

`that we can apply the Löwenhein-Skolem theorem saying that there is`

`enumerable model of ZF. A model is just an informal set satisfying all`

`the axioms of ZF. Furthermore it can be shown that there is transitive`

`enumerable model M, and this means that if A belongs to M, then A is`

`included in M: all elements of A are element of the model M. But ZF`

`proves the existence of a non countable set A. So it looks like if`

`there is a non countable set included in a countable set! The solution`

`of that paradox is that the non countable set is just non countable`

`when seen in the model M. It means that the bijection between A and M`

`is not itself a set in M. The bijection go outside of the universe M.`

`This shows also that the notion of cardinality is relative to the`

`model in which we are working. M is countable, but as seen in the`

`model M it is bigger than the uncountable.`

`For a mechanist, this should not be so much more astonishing than the`

`fact that a finite machine can conceive an infinite set, or that a`

`little brain can conceive a vast, even infinite, universe.`

`It is hard to explain more without getting more technical. I suggest`

`you search on "Skolem Paradox". It illustrates that phenomenon very`

`well. But the "real" explanation is that the first person is directly`

`defined by an evocation to the non arithmetical notion of truth. The`

`inside first person view of arithmetic is not arithmetic. In a sense`

`the first person associated to a machine is not a machine, from its`

`personal point of view. It explains also why it is necessary to make a`

`leap of faith when saying "yes" to a digitalist surgeon, and why`

`mechanism is intuitively unbelievable by machines.`

Bruno

How is it possible to have an outside view of the ultimate reality?By assuming mechanism. If mechanism fails the UDA test, we willknow that arithmetical truth is too big or to small. Plausibly toosmall (for the reason that we can already conceive, apparently,arithmetical truth (but an ultrafinitist could contest this).Church thesis rehabilitates the Pythagorean Neoplatonist theology.The relation between the numbers (the true arithmetical relation orpropositions) defines, by the first person indeterminacy, adifferentiating flux of consciousness, which, by the non awarenessof the basic UD execution time (defined by those relations),filters *in the limit* the possible and sharable infiniteconsistent histories.The reason why you are conscious here and now is rather easy. Itdepends only on the truth of one sigma_1 arithmetical sentence(asserting the existence of your computational state relatively tosome universal number). But there are infinitely many equivalentsentences, both with the same universal numbers and with differentuniversal numbers. Most relations involve pretty big numbers.What the first person indeterminacy makes complex is the relationbetween 'here and now', and 'there and then'. This, unfortunatelydepends on that infinity of universal (among others) numbers, andall those sigma_1 propositions, and actually all their proofs,which define the relative measure.Below your substitution level, an infinity of universal numbercompete, for proving your existence, together with all randomoracles.Fortunately, the constraints of self-referential correctness forthe Löbian observers are enough to shed some light on this, and tomake quite plausible that the bottom physical reality will appearlinear and extremely symmetrical, then self-observation break thesymmetries from the point of view of the self-observing machines.You can see this as Everett spirit applied to arithmetic. Everettembeds the physicist in the quantum reality. DM embeds themathematician in the arithmetical reality.I choose the numbers, but any finite combinatorial objects woulddo. Physics, and none of the internal views depends on the initialchoice, except perhaps(*) some diagonal systems build to lead toanomalous measure.Bruno (*) a conjecture by Jacques Bailhache on this list.--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-l...@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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