On 2/6/2012 1:50 AM, Bruno Marchal wrote:
On 05 Feb 2012, at 21:32, meekerdb wrote:
On 2/5/2012 8:19 AM, Bruno Marchal wrote:
No. All universal numbers can interpret a number as a function on quantities, or as
properties on quantities, which are not quantities themselves. Universal numbers can
also transform, or interpret numbers as transformation of transformation, properties
of properties, up in the constructive transfinite, etc.
When the quantities can add and multiply, soon their attributes are beyond all
quantities, and Löbian quantities are arguably already knowing that about themselves.
I don't understand this. Maybe I don't know what universal number is. I thought it
was a number whose representation in digits was such that every number appeared in the
representation. But I don't understand how such number does things: transform,
interpret,...
Let phi_i be an enumeration of the (partial and total) computable functions
from N to N.
Let <x,y> be a bijection from NXN to N.
A universal number u is a number u such that, for all x and y, we have phi_u(<x,y>) =
phi_x(y).
The equality means that the LHS and RHS are either both defined and equal, or both
undefined.
Thanks. So it is not literally that the number does things, it just picks out the
function that is universal for a given bijection and a given enumeration of the functions.
Brent
u, applied on x and y simulate the machine x on the input y. u is called the computer, x
the program (the machine to be emulated), and y is the datum/data. u interpret x as a
machine, and it simulates x behavior on the input y.
You can see it as the number-code of a universal machine or programming language
interpreter.
u depends on the choice of the bijection and of the phi_i base, but if you choose (N, +,
*) as a universal system, you can make it intrinsic, and for any bijection, you will
have different but equivalent universal numbers. This is not a problem because we have
to consider *all* universal numbers to retrieve the physics and psychology of machines
(this will include all such bijection).
Bruno
http://iridia.ulb.ac.be/~marchal/ <http://iridia.ulb.ac.be/%7Emarchal/>
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