Robert Shaw wrote:
>
> The lower bound of the ste of distances between all pairs of
> numbers in the set.
>
> If for any epsilion>0 there are two points in the set a and b
> such that |a-b|<epsilon then we can say the minimum distance
> between points in the set is zero. Points in the set can be
> arbitarily close together.
>
Ok. But what is the purpose of this concept? We agree that
if X is uncountable then Acc(X) is uncountable; if
X is countable then we can't say anything about Acc(X).
This "minimum distance between points in the set" doesn't
even guarantee [when it's zero] that Acc(X) is non-empty,
as, for example, for
X = { 1, 1 + 1/2, 1 + 1/2 + 1/3, ... }
>
> Well, we've implicitly idealised to infinite precision
> arithmetic anyway or we could only ever produce rationals.
>
Yes - and not all of them :-)
> We can ignore errors produced
> by numbers out of range, such as 3!!!!, in the same way.
>
But I don't think we should ignore _error_ as the output
of the functions, if we want to apply a rigorous
mathematical treatment to the problem.
> (...)it's interesting enough just to consider what
> type of positive reals we can get this way, either from this set
> of functions, or from arbitary finite sets of functions defined
> over some interesting domain.
>
But this is the problem that we are trying to discuss. Or,
there are two problems: which numbers can be constructed
in an exact way, and which numbers are the accumulation
points of the set of numbers that can be constructed in an
exact way. The first set is enumerable, maybe finite. The
second set might be anything.
If "(" and ")" and the numeric keypad are allowed, then,
using an ideal calculator with just arithmetic and power
functions, we know we can't construct any transcendental
number, or even the roots of some 5-th degree polynomials.
Also, I think some numbers like the solution of cos(x) = x
can't be constructed using *all* buttoms of the calculator,
but I don't know if this has been proven or if it's a
conjecture.
But, anyway, the solution to these problems must come from
"let f1, f2, ... be functions from K [ x K] -> K ...", and,
IMHO, K must be R U { _error_ }
Alberto Monteiro
