On 12 Mar 2011, at 19:17, Brent Meeker wrote:

On 3/12/2011 1:48 AM, Bruno Marchal wrote: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 isimplicit in any textbook on theoretical computer science, or ofrecursion theory (like books by Cutland, or Rogers, or Boolos andJeffrey, ...).Fix any universal system, for example numbers+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 mechanicallyextract x 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 RobinsonArithmetic prover. I will say that a number u is universal if RAcan prove the (purely arithmetical) relation phi_u(<x,y>) =phi_x(y).The notion is not entirely intrinsic (so to be universal is notlike to be prime), but this is not important because from themachine's point of view, all universal numbers have to be takeninto 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, eachuniversal numbers will have many syntactical variants due to theprovable existence of different coding scheme. This does play arole in the measure problem a priori. But this is true for yourbrain too (clearly so 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 oranother. So what does "taking into account" mean?Not every natural number will be universal. Once you have chooseyour initial system, only a sparse set of numbers will beuniversal, and the computations will rely on the relations betweenthe numbers. You are not supposed to change the initial system. Thelaws of mind and the laws of matter does not depend on the choiceof the theory (initial system), but you cannot change the theory,once working in the theory. To say that all universal numbers haveto be taken into account means that you have to take into accountall the relations which constitute computations in the initialtheory. The observer cannot distinguish them from its first personpoint of view. The measure does not depend on the codings, but doesdepend on the existence 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 usea universal 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 asa system of diophantine polynomials(*). I will say that a number Nuis universal 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 ofbeliefs, themselves represented by RE set W_i, etc. The same istrue for "interaction" between Löbian machine and environments(computable or computable with oracle), etc.For any choice of representation, the number relation organizesitself into a well defined mathematical structure with the sameinternal hypostases.Here is a universal system of diophantine polynomial, taken in apaper by 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 withRA would be much longer and tedious, but the principle is the same.Once the initial universal system is chosen, there is no morearbitrariness in the notion of universal numbers, computations,etc. Except for different internal codings, but this is the pricefor comp. I can change your brain in many ways, but I have to keepthe relations between your constituents intact, making your brain anon relative notion. A brain is not intrinsic, but is not arbitraryeither.In the universal system of diophantine polynomials below, there isan infinity of number Nu coding the primes numbers, or the evennumbers, or the universal numbers, etc. If all numbers code precisethings, it is false that precise things can be coded by arbitrarynumbers.(*) (Numbers are named by single capital letters, or by singlecapital letters + one non capital letter. In the first line Nu, Z,U, Y, are integers.They are all integers, aren't they? So am I to understand Z, U, andY as parameters which can be chosen and each choice will pick out aset of numbers, such as the primes, the evens,..., Goldbach pairs?

Yes. Even non negative integers in that particular example.

Then X belongs to W_Nu is equivalent withSo the solutions, X, will be a set of integers which, in our chosencoding scheme, express some recursively enumerable set of numbers?

Yes.

`The set of numbers X will constitute such a W_Nu, and is a recursively`

`enumerable set of all solutions of the system of diophantine`

`polynomials for some choice of the parameters Z, U, Y. And all W_i can`

`be retrieved by chosing well Z, U, Y.`

`A number Nu can be said to be a universal numbers if Nu = ((ZUY)^2 +`

`U)^2 + Y, + the other polynoms, and W_nu is a universal recursively`

`enumerable set (a creative set, a m-complete set or a sigma_1 complete`

`set, those are equivalent notion).`

`There will be an infinity of them, but few numbers, in proportion,`

`will be universal. Like there is an infinity of codes for a LISP`

`interpreter, but few programs are LISP interpreter.`

`It is hard to believe that a super growing function like y =`

`x^(x^(x^(x^ ...x) with x occurrence of x (which grows quicker than all`

`exponential functions!) can be represented by a polynomial. But this`

`has been proved and is a direct consequence, of course, of the`

`existence of universal diophantine polynomial. There is also parameter`

`Z, U, Y coding the current state of our galaxy, or coding the quantum`

`vacuum ... Those are (sophisticate) example of universal numbers.`

`Note that the degree of the universal polynomial below is 5^60. We can`

`reduce the degree to four, but then we will need 58 variables.`

`Note that if the variables are real numbers, we get a decidable, even`

`finite, and thus non universal, set of solutions. The universality`

`comes from the digitality, or from the natural numbers. The natural`

`numbers behave in an infinitely more complex way than the real`

`numbers, and reminds us of the non triviality of what is a`

`computation. That can perhaps be used to mitigate the idea that a rock`

`can think, because I can imagine that point particles (in quantum non`

`relativistic physics for example) can simulate the real-number version`

`of that polynomial, but I am less sure for the diophantine (integers)`

`version.`

`Note also that, although the first order theory of rational numbers is`

`Turing universal, it is an open problem if there is a universal`

`polynomial on the rational numbers. Another open problem is the`

`existence of a cubic universal diophantine polynomial. We know that`

`there is no quadratic universal polynomial, though.`

Bruno

BrentNu = ((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 lotof place and variables. The solutions of that equations provide auniversal dovetailing, notably because computations can be seen asRE relations. For further reference, I will call that system JJ(for James Jones). JJ, like RA, is an acceptable theory ofeverything, at the ontological level.Only the left (terrestrial) hypostases can be named in that system,the right hypostases, the divine one, are, in the general casepurely epistemological and cannot be named in the system, like theuniversal soul itself S4Grz. Machines can point on it onlyindirectly, but can study, nevertheless, the complete theory forsimpler Löbian machines than themselves. So JJ provides theontology, but the internal epistemology necessarily escapes theontological TOE (whatever the initial theory choice). This explainthat you can *see* consciousness there, no more than you can seeconsciousness 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 theone ruled 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 themwhen extended in first order modal logic. Philosophy of mind, withcomp, is intrinsically very complex mathematics. The contrary wouldhave been astonishing, to be sure.To sum up: don't confuse the flexibility of the codings with thenon arbitrariness of the relations we obtain between those numbersin any initial theory chosen.Note that JJ is simpler than RA. It has fewer assumption. It doesnot need 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/--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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