> SBJ: Information entropy of physical fundamental constants
> The fundamental constant can be measured increasingly accurate, it
> does not seem (for me) that the  repetitive pattern of rational
> numbers after some number of digits may take place; 

Physical measurements are always relative, i.e. one quantify is measured in 
units of another quantity.  It is generally thought that there is a smallest 
possible unit, the Planck scale, so all physical measurements will be rational 
numbers (integers in Planck units).

>if it is the case
> then there is not enough "room" in the universe / multiverse to
> accommodate such information as exact representation of fundamental
> constant - just in principle, there is no way to have it exact as
> there is no way to write down an exact arbitrary irrational number and
> it is not a technical limitation it is a fundamental limitation unless
> it may be represented as a rational number ;o). 

There is no problem writing down irrational numbers: sqrt(2), pi,... See 
to it.  ;-)

Of course from an information standpoint you want to know their bits.  But it 
also easy to write down a quite short program that will compute whatever bit 
want to know for those irrational numbers.  But you are right that for almost 
all real numbers is impossible to give them a finite representation.  But why 
believe in those numbers anyway, they are convenient fictions.

>Information Entropy
> can be measured as an average number of bits per symbol/digit encoded
> by rank-0 context model + entropy encoder (let say arithmetic
> encoder). Therefore, there are two distinct possibilities: entropy
> equals zero or Log2(10) (for decimal representation) or simply: ZERO
> or NON-ZERO. I have my ideas how NON-ZERO case may workout but I'm
> interested to listen others opinions.

Most cosmogonies assume the (microscopic) entropy of the universe is zero.  It 
started at the Planck scale, where there is room for at most one bit and since 
QM insists on unitary evolution the entropy cannot change (as measured at the 
Planck scale).  The increase in entropy we see is due to our coarse graining, 
as Bruno would say, "above our substitution level".  It is impossible to 
to use the negative information to get back to local zero because the expansion 
of the universe has carried the correlations beyond the relativistic horizon. 
At least that's the common theory.


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