On Mon, May 23, 2005 at 11:17:04PM +0100, Patrick Leahy wrote: > And another mathematical query for you or anyone on the list: > > I've overlooked until now the fact that mathematical physics restricts > itself to (almost-everywhere) differentiable functions of the continuum. > What is the cardinality of the set of such functions? I rather suspect > that they are denumerable, hence exactly representable by UTM programs. > Perhaps this is what Russell Standish meant. >

I noticed a couple of people have already responded to this with the obvious, but you can consider the set of analytic functions (which tend to be popular with physicists), which are representable as a Taylor series, so can be represented by a countable set of real coefficients. If you further restrict the coefficients to be computable (namely the limit of some well defined series) then the cardinality of your set is countable. Not sure if this observation translates to the bigger set of almost everywhere differentiable functions. Also, can one create a consistent theory of differential calculus based on a field of computable numbers (it must be a field, because it is bigger than the set of rationals). The reason I persist in this notion is that the set of all descriptions has the interesting property of having zero information. This set does have cardinality c, however. Cheers -- *PS: A number of people ask me about the attachment to my email, which is of type "application/pgp-signature". Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 8308 3119 (mobile) Mathematics 0425 253119 (") UNSW SYDNEY 2052 [EMAIL PROTECTED] Australia http://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 ----------------------------------------------------------------------------

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