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
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Received on Mon May 23 2005 - 20:09:31 PDT