Robert Shaw wrote:
>
> Actually, this is basically the continued fraction expansion.
>
I knew O:-)
>
> That means any real can be produced by some infinite
> sequence of button presses, which wasn't self-evident.
>
This is another, similar, problem. Which numbers can be
*approximated* by an infinite sequence of two funcions?
For example, if you just have x -> x! and x -> sqrt(x),
can you get as close as you want to any number? This is
an extreme case, because you can't ()! non-integer numbers
[calling x! = Gamma(x + 1) is cheating!!!]
> Which numbers can be reached with a finite sequence is
> a harder question, but I don't think it's a well known set.
>
I think the toughest part is proving that some number
*can't* be reached. I mean, we know [countability arguments]
that such numbers exists, but can we point to some number,
[say, "the solution of x = cos(x)"] and *prove* that it
can't be reached?
> The largest standard countable set is the algebraic numbers,
> the other infinite countable sets being subsets of that.
>
"Largest standard"?
> For the set of constructable numbers to be that set you'd
> have to be able to take nth roots of any number with these
> operations (and solve more general polynomials) but thats
> equivalent to deviding or multiplying by an arbitary integer.
> I don't see any way of doing that so I suspect the constructable
> set is not the algebraic numbers, but I don't see any proof.
>
Yep, I can't see how to get n * x from x either, because
there's no recursive operation to get (n + 1) * x from
n * x short of rebuilding x again - and we can't store
stuff.
> The constructable set does include all quadratic and cubic
> irrationals, e, pi, and the log of every integer, so it includes
> most of the best known transcendentals.
>
Can you claim that some number is *not* constructable?
Alberto Monteiro
