Bruno,
In one sense those examples are things for which (finite) reasoning
fails, but I would still say that they are governed by (finite) rules
and possess a (finite) description-- the problem is "merely" that it
takes infinite amounts of time to derive the consequences of those
rules/descriptions.
--Abram
On Fri, Dec 26, 2008 at 11:49 AM, Bruno Marchal <marchal.domain.name.hidden> wrote:
>
>
> 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
> machines.
> The set of descriptions (in any universal language) of any non trivial
> machines.
> 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-
> effectivity.
> 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
> set.
>
> Bruno
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
>
> >
>
--
Abram Demski
Public address: abram-demski.domain.name.hidden
Public archive: http://groups.google.com/group/abram-demski
Private address: abramdemski.domain.name.hidden
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Received on Fri Dec 26 2008 - 14:24:19 PST