Le 22-août-06, à 08:36, Tom Caylor a écrit :

> I believe that we are finite, but as I said in the "computationalsim
> and supervenience" thread, it doesn't seem that this is a strong enough
> statement to be useful in a TOE.  It seems that you cannot have YD
> without CT, but if true I would leave Bruno to explain exactly why.
>

I am not sure I have said that YD needs CT. CT is needed to use the 
informal "digital" instead of the "turing", "java" "python" seemingly 
restriction.
For someone not believing in CT, "digital" could have a wider meaning 
than "turing emulable".
Now CT needs AR. CT is equivalent with the statement that all universal 
digital machine can emulate each other. To make this precise (or just 
to define universal machine/number) you need to believe in numbers. 
(But just in the usual sense of any number theorist).
Recursion theory is really a branch of number theory, although few 
number theorist would accept this joyfully. Well a notable exception is 
Yuri Manin. He wrote a beautiful (but advanced) book on number theory 
which has a impressive chapter on recursion theory (the theory of the 
Wi and Fi).

Well, thanks to Matiyasevitch, few number theorist would seriously 
argue that recursion theory has nothing to say about numbers in the 
sense that you need recursion theory and even Church thesis to say that 
the 10th Hilbert problem has been solved negatively. By using compute 
science (alias elementary recursion theory), Matiyasevitch has indeed 
shown that there is no algorithm (this makes sense only with CT) for 
the resolution of diophantine equations (polynomials with integers 
coefficient and with integers or natural number for the variables or 
unknowns).

Bruno

http://iridia.ulb.ac.be/~marchal/


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