On 25 Dec 2008, at 22:27, Kim Jones wrote:

> On 26/12/2008, at 5:23 AM, Bruno Marchal wrote:
>> On 25 Dec 2008, at 08:05, Abram Demski wrote:
>>> Bruno,
>>> I agree with Gunther about the two types of machine. The broader
>>> machine is any system that can be logically described-- a system  
>>> that
>>> is governed by rules and has a definite description.
>> Then Church thesis entails it is not broader, unless you mean that
>> the rules are not effective.
> I might be missing something here, but somebody please give an example
> of a system that is NOT governed by rules and possesses NO definite
> description.

Arithmetical truth. That is, the set of all true sentences of  
elementary arithmetic.
The set of Gödel number, or description of never stopping programs or  
The set of descriptions (in any universal language) of any non trivial  
At the first order level: all the arithmetical hypostases.
Sigma_2 truth, Sigma_3 truth, Sigma_4 truth, Sigma_5 truth, Sigma_6  
truth,  etc. (the union of which gives arithmetical truth)
Analytical truth (far beyond arithmetical truth).
Mathematical Truth (if that exists).

Kim, all those exemples provide well defined set of objects, (except  
the last one) but there is no way to generate them by any machine, nor  
can we axiomatize them in any effective way. No effective complete  
"Theory" for any of them.

Alas, there is a need of some math to prove this. If you are patient,  
when we get the seven step of UDA, I will have to give you at least a  
tool (diagonalization) capable of easily showing the existence and the  
non effectivity of those non mechanical mathematical realities.

It is needed to be more precise on "effectivity" to discover the non- 
Mechanism is not a reductionism, (as I explain often to John Mikes)  
because Universal machines behaviors depends on those non effective  
things. Creation and life appears on the border between the computable  
and the non computable. It is similar to the border of the Mandelbrot  



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