Bruno Marchal wrote:
> 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

I agree with the importance of recursion theory.  By the way I got the
book by Cutland.


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