On 3/9/2011 9:14 AM, Bruno Marchal wrote:

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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. Couldyou 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 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 mechanically extract xand 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 Godel`

`numbering scheme that idenitfies a number u with a program is not`

`something intrinsic to arithmetic; it is a mapping we create. Right?`

`But then what does it mean to say "all universal numbers have to taken`

`into account"? If I understand correctly *every* natural number is a`

`universal number, in some numbering scheme or another. So what does`

`"taking into account" mean?`

Brent

With that respect, here, mind theorist have an easier work thancomputer scientist which search intrinsic notion of universality. Wedon't need that, because the personal Löbian machine and theirhypostases does not depend on the initial choice, neither of thecomputable bijection, nor of the "first universal" system.To put it more simply: a universal number is the Gödel number of thecode of a universal system (a computer, or a general purpose computer(in french: an 'ordinateur'), or a 'programming language interpreter').OK? Ask for more if needed. >> Bruno http://iridia.ulb.ac.be/~marchal/

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