On 06 Feb 2012, at 19:34, meekerdb wrote:
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.
You are right. Technically we could bypass the bijection, and use only
function with one argument, but this leads to the combinators (or
numbers with some other operations). But a number per se do nothing.
It needs a universal numbers to be interpreted, or ... the fixed
universal base and in that case the * and + laws are enough, so that
the choice of the universal (Sigma_1 complete) initial system endows
the numbers with a "natural operational interpretation" by that
universal basic system. From inside, the numbers will still not know
the difference between a base (UD-like) computations, and any higher
level one (and for the physics he has to take them all into account).
Bruno
http://iridia.ulb.ac.be/~marchal/
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