> On 3 Mar 2019, at 03:27, John Clark <johnkcl...@gmail.com> wrote:
> 
> 
> On Fri, Mar 1, 2019 at 4:23 PM Lawrence Crowell 
> <goldenfieldquaterni...@gmail.com <mailto:goldenfieldquaterni...@gmail.com>> 
> wrote:
> 
> > There are numbers that have no description in a practical sense. The 
> > numbers 10^{10^{10^{10}}} and 10^{10^{10^{10^{10}}}} have a vast number of 
> > numbers that have no description with any information theoretic sense.
> 
> The 8000th Busy Beaver Number can be named but not calculated even 
> theoretically,


The busy beaver function is not computable, but on each individual n, it is 
computable theoretically, of course, not practically, as there is no room in 
the observable universe to even write the result.

The 8000h BB number is well defined, so it is a (finite) number, and so 
youthere exist a finite program computing it: just “write that numbers”. All 
initial segment of non computable function are computable with a look-up table. 
(All finite function are computable).






> but most Real Numbers can't even be uniquely named with ASCII characters, not 
> even with an infinite number of them.  


Yes, like the Total computable functions. Turing defined a computable real 
number par the total computable function giving its decimal, or its 
approximation. That is a good simplifying definition, but it makes, as Turing 
eventually realise, addition and multiplication not computable. The “modern” 
definition of computable real numbers add some condition of computable modulus 
of convergence to remedy. Note that if there is a miraculous “Church’s thesis” 
for the computable functions on finite objects, there is no equivalent thesis 
for the reals, although there are relations with the analytical hierarchy. 
People can Google on elementary, hyperelementary, projective Borel …” to get 
more information of this. 

Bruno



> 
> John K Clark
> 
> 
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