On Mon, 23 May 2005, Hal Finney wrote:
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.
The cardinality of such functions is c, the same as the continuum.
The existence of the constant functions alone shows that it is at least c,
and my understanding is that continuous, let alone differentiable, functions
have cardinality no more than c.
Oops, mea culpa. I said that wrong. What I meant was, what is the
cardinality of the data needed to specify *one* continuous function of the
continuum. E.g. for constant functions it is blatantly aleph-null.
Similarly for any function expressible as a finite-length formula in which
some terms stand for reals.