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.
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).
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