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

I agree with the importance of recursion theory. By the way I got the book by Cutland. Tom --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~----------~----~----~----~------~----~------~--~---