On 3/12/2011 1:48 AM, Bruno Marchal wrote:

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On 11 Mar 2011, at 19:53, Brent Meeker wrote:On 3/9/2011 9:14 AM, Bruno Marchal wrote:On 08 Mar 2011, at 20:11, Brent Meeker wrote:On 3/8/2011 10:41 AM, Bruno Marchal wrote:We could start with lambda terms, or combinators instead. Acomputation (of phi_4(5) is just a sequencephi_4^0 (5) phi_4^1 (5) phi_4^2 (5) phi_4^3 (5) phi_4^4 (5)phi_4^5 (5) phi_4^6 (5) ..._4 is the program. 5 is the input. ^i is the ith step (i = 0, 1,2, ...), and phi refers implicitly to a universal number.Bruno, I don't think I understand what a universal number is.Could you point me to an explication.The expression "universal numbers" is mine, but the idea is implicitin any textbook on theoretical computer science, or of recursiontheory (like books by Cutland, or Rogers, or Boolos and Jeffrey, ...).Fix any universal system, for examplenumbers+addition+multiplication, or LISP programs.You can enumerate the programs: P_0, P_1, P_2, ... So that you can enumerate the corresponding phi_i phi_0, phi_1, phi_2, ...Take a computable bijection between NXN and N, so that couples ofnumbers <x,y> are code by numbers, and you can mechanically extractx and y from <x,y>Then u is a universal number if for all x and y you have thatphi_u(<x,y>) = phi_x(y).In practice x is called program, and y is called the input.Now, I use, as fixed initial universal system, a Robinson Arithmeticprover. I will say that a number u is universal if RA can prove the(purely arithmetical) relation phi_u(<x,y>) = phi_x(y).The notion is not entirely intrinsic (so to be universal is not liketo be prime), but this is not important because from the machine'spoint of view, all universal numbers have to be taken into account.By "not intrinsic" I assume you refer to the fact that the Godelnumbering scheme that idenitfies a number u with a program is notsomething intrinsic to arithmetic; it is a mapping we create. Right?Partially right. If you start from Robinson arithmetic, each universalnumbers will have many syntactical variants due to the provableexistence of different coding scheme. This does play a role in themeasure problem a priori. But this is true for your brain too (clearlyso with comp).But then what does it mean to say "all universal numbers have totaken into account"? If I understand correctly *every* naturalnumber is a universal number, in some numbering scheme or another.So what does "taking into account" mean?Not every natural number will be universal. Once you have choose yourinitial system, only a sparse set of numbers will be universal, andthe computations will rely on the relations between the numbers. Youare not supposed to change the initial system. The laws of mind andthe laws of matter does not depend on the choice of the theory(initial system), but you cannot change the theory, once working inthe theory. To say that all universal numbers have to be taken intoaccount means that you have to take into account all the relationswhich constitute computations in the initial theory. The observercannot distinguish them from its first person point of view. Themeasure does not depend on the codings, but does depend on theexistence of infinitely many coding schemes.

OK. That's what I thought, but I wanted to be sure.

Let me be completely specific, and instead of using RA, let us use auniversal system of diophantine polynomials:The phi_i are the partial computable functions. The W_i are theirdomain, and constitutes the recursively enumerable sets.The Turing universal number relation (X is in W_Nu) can be coded as asystem of diophantine polynomials(*). I will say that a number Nu isuniversal if W_Nu is the graph of a universal computable function.The detail of the polynomials is not important. I want just to showthat such numbers are very specific. Then you can representcomputations by their recursively enumerable sequence of steps, etc.Löbian machine can likewise be represented by their set of beliefs,themselves represented by RE set W_i, etc. The same is true for"interaction" between Löbian machine and environments (computable orcomputable with oracle), etc.For any choice of representation, the number relation organizes itselfinto a well defined mathematical structure with the same internalhypostases.Here is a universal system of diophantine polynomial, taken in a paperby James JonesJames P. Jones, Universal Diophantine Equation, The Journal ofSymbolic Logic, Vol. 47, No. 3. (Sept. 1982), pp. 549-571.To define universal numbers here is very short. To do the same with RAwould be much longer and tedious, but the principle is the same. Oncethe initial universal system is chosen, there is no more arbitrarinessin the notion of universal numbers, computations, etc. Except fordifferent internal codings, but this is the price for comp. I canchange your brain in many ways, but I have to keep the relationsbetween your constituents intact, making your brain a non relativenotion. A brain is not intrinsic, but is not arbitrary either.In the universal system of diophantine polynomials below, there is aninfinity of number Nu coding the primes numbers, or the even numbers,or the universal numbers, etc. If all numbers code precise things, itis false that precise things can be coded by arbitrary numbers.(*) (Numbers are named by single capital letters, or by single capitalletters + one non capital letter. In the first line Nu, Z, U, Y, areintegers.

`They are all integers, aren't they? So am I to understand Z, U, and Y`

`as parameters which can be chosen and each choice will pick out a set of`

`numbers, such as the primes, the evens,..., Goldbach pairs?`

Then X belongs to W_Nu is equivalent with

`So the solutions, X, will be a set of integers which, in our chosen`

`coding scheme, express some recursively enumerable set of numbers?`

Brent

Nu = ((ZUY)^2 + U)^2 + Y ELG^2 + Al = (B - XY)Q^2 Qu = B^(5^60) La + Qu^4 = 1 + LaB^5 Th + 2Z = B^5 L = U + TTh E = Y + MTh N = Q^16R = [G + EQ^3 + LQ^5 + (2(E - ZLa)(1 + XB^5 + G)^4 + LaB^5 + +LaB^5Q^4)Q^4](N^2 -N)+ [Q^3 -BL + L + ThLaQ^3 + (B^5 - 2)Q^5] (N^2 - 1) P = 2W(S^2)(R^2)N^2 (P^2)K^2 - K^2 + 1 = Ta^2 4(c - KSN^2)^2 + Et = K^2 K = R + 1 + HP - H A = (WN^2 + 1)RSN^2 C = 2R + 1 Ph D = BW + CA -2C + 4AGa -5Ga D^2 = (A^2 - 1)C^2 + 1 F^2 = (A^2 - 1)(I^2)C^4 + 1 (D + OF)^2 = ((A + F^2(D^2 - A^2))^2 - 1)(2R + 1 + JC)^2 + 1Z, U, Y codes just the parameter Nu. The other "letters" are thevariables.We could transform this into one giant universal polynomial, and wecould manage its degree to be equal to 4. But that would take a lot ofplace and variables. The solutions of that equations provide auniversal dovetailing, notably because computations can be seen as RErelations. For further reference, I will call that system JJ (forJames Jones). JJ, like RA, is an acceptable theory of everything, atthe ontological level.Only the left (terrestrial) hypostases can be named in that system,the right hypostases, the divine one, are, in the general case purelyepistemological and cannot be named in the system, like the universalsoul itself S4Grz. Machines can point on it only indirectly, but canstudy, nevertheless, the complete theory for simpler Löbian machinesthan themselves. So JJ provides the ontology, but the internalepistemology necessarily escapes the ontological TOE (whatever theinitial theory choice). This explain that you can *see* consciousnessthere, no more than you can see consciousness by dissecting a brain.I recall the 8 hypostases (three of them splits according to the G/G*splitting)p Bp Bp & p Bp & Dp Bp a Dp & pThe "left one" are the one ruled by G, and the right one are the oneruled by G*. Which corresponds the terrestrial and the divinerespectively in the arithmetical interpretation of Plotinus. Thosewith (& p) are not solution of the JJ equations, nor any of them whenextended in first order modal logic. Philosophy of mind, with comp, isintrinsically very complex mathematics. The contrary would have beenastonishing, to be sure.To sum up: don't confuse the flexibility of the codings with the nonarbitrariness of the relations we obtain between those numbers in anyinitial theory chosen.Note that JJ is simpler than RA. It has fewer assumption. It does notneed logics, but only equality axioms, and a the existentialquantifier (or the distinction variable/parameter). It is theequivalent of the other "everthing ontological theory" based on thecombinators:((K x) y) = x (((S x) y) z) = ((x z) (y z))Which is quite much readable and shorter! In that theory you mightunderstand that there will be an infinity of universal combinators(which will be very long!), but to be universal will not be anarbitrary notion. Non-intrinsic does not mean arbitrary.Bruno http://iridia.ulb.ac.be/~marchal/

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