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 complete description of reality or a complete description of everything that exists.That does not exist. Arithmetical truth is already not recursively enumerable.If a complete description of arithmetical truth is not possible, what exactly are we talking about?

`We talk about the set of all true arithmetical statements; written in`

`the (first order logic) language of arithmetic.`

`That set contains the verifiable atomic formula like the usual taught`

`in school: 0 + 1 = 1, 45 + 9 = 54, 3*7 + 3 = 24, etc. Where 3`

`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 universal`

`number is sigma_1. Then the true pi_1 sentences like Ax(x = 0 V not(x`

`= 0)).`

`Here we can already show that for any (proving/believing) machine`

`there is an arithmetical pi_1 proposition which is true but not`

`provable by that machine. Note that the Riemann hypothesis can be`

`shown to be equivalent with an arithmetical pi_1 sentence, and is an`

`example of a pi_1 sentence that we suspect to be true but that we can`

`still not prove.`

`Then the sigma_2 sentences, equivalent to a proposition with the shape`

`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 are beyond the`

`human mind.`

`Arithmetical truth is the set of all true propositions in arithmetic.`

`That set is not recursively enumerable (cannot be defined by a sigma_1`

`proposition, and actually is not arithmetical (cannot be defined by`

`*any* arithmetical proposition!). That follows from Gödel and Tarski.`

`We, humans, have a rather good intuition of what is a true`

`arithmetical sentence, independently of the fact that we have to`

`recognize that it can be quite tricky to decide if this or that`

`arithmetical proposition is true or not.`

Then one *might* consider this program which generates a TOE to arbitrary precision to be "the" TOE, a compression of an infinitelylong 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 logical structure which is a tuple consisting of an underlying set, a set of distinguished constants (like zero), functions (like successor), and relations (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 infinite and`

`non algorithmically generable.`

`It is just absolutely undecidable that there is anything more needed`

`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 that consciousness is`

`preserved) then the inside "knowledge" view of arithmetical truth will`

`be FAR BIGGER than arithmetical truth. Indeed, it corresponds`

`plausibly to George Levy first person plenitude. That things is much`

`less definable (if I can say) than arithmetical truth. There is here`

`an analog of Skolem paradox, with the existence of a structure which`

`looks small from outside and is immensely big from inside.`

How is it possible to have an outside view of the ultimate reality?

`By assuming mechanism. If mechanism fails the UDA test, we will know`

`that arithmetical truth is too big or to small. Plausibly too small`

`(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 or`

`propositions) defines, by the first person indeterminacy, a`

`differentiating flux of consciousness, which, by the non awareness of`

`the basic UD execution time (defined by those relations), filters *in`

`the limit* the possible and sharable infinite consistent histories.`

`The reason why you are conscious here and now is rather easy. It`

`depends only on the truth of one sigma_1 arithmetical sentence`

`(asserting the existence of your computational state relatively to`

`some universal number). But there are infinitely many equivalent`

`sentences, both with the same universal numbers and with different`

`universal numbers. Most relations involve pretty big numbers.`

`What the first person indeterminacy makes complex is the relation`

`between 'here and now', and 'there and then'. This, unfortunately`

`depends on that infinity of universal (among others) numbers, and all`

`those sigma_1 propositions, and actually all their proofs, which`

`define the relative measure.`

`Below your substitution level, an infinity of universal number`

`compete, for proving your existence, together with all random oracles.`

`Fortunately, the constraints of self-referential correctness for the`

`Löbian observers are enough to shed some light on this, and to make`

`quite plausible that the bottom physical reality will appear linear`

`and extremely symmetrical, then self-observation break the symmetries`

`from the point of view of the self-observing machines.`

`You can see this as Everett spirit applied to arithmetic. Everett`

`embeds the physicist in the quantum reality. DM embeds the`

`mathematician in the arithmetical reality.`

`I choose the numbers, but any finite combinatorial objects would do.`

`Physics, and none of the internal views depends on the initial choice,`

`except perhaps(*) some diagonal systems build to lead to anomalous`

`measure.`

Bruno (*) a conjecture by Jacques Bailhache on this list.

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