"Alberto Monteiro" wrote
> 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?
Limited. I was thinking about what si needed for the set
to be dense, but this concept is a much weaker condition.
> 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, ... }
>
If the minimum distance isn't zero then the points in the
set divide the reals into a countable union of intervals.
Other than that, it doesn't say much useful.
> >
> > 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.
>
There are two types of errors.
One comes from the finite memory of the calculator, which
restricts it to rationals within a certain range. This restriction isn't
a inherent property of the calculator functions, but depends
on the calclator. It's not an interesting restriction.
The other type of error comes from taking a function outside
its domain, acos(2) or 0.3! These errors are inherent in the
choice of function, not physical artefacts or the particular
calculator. Errors produced this way are interesting, and should
not be ignored.
> > (...)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.
>
We know the second set is uncountable, and includes the
rationals.
Their are five sets involved, so for clarity they need names.
We've got the set of calculator functions Ca={cos,sin,tan,exp etc)
Second, the set of all finite sequences of calculator functions, S.
S is countable, with elements s of the form (fn,....f3,f2,f1) with
each fm being an element of Ca.
We have a mapping G from S to K=R U{_error_} given
by G(s) =fn(...f3(f2(f1(0)..)
Define fn(_error_)=_error_ for all fn in Ca
G(S) is the set of all finitely constructable number starting with 0.
For arbitary set Ca G(S) is not automatically countable. If
c={1/x,x^2}then G(S)={0,_error_}
For this C we know Q is a subset of G(S), as you proved,
so G(S) is countably infinite and dense.
We don't know what G(S) is, though we think the
algebraic numbers A are not a subset of G(S)
Next we've got the set Si of all infinite sequences of functions
in Ca. By Cantor's diagonal argument we know Si is
an uncountable set if at least one pair of functions in Ca does
not commute.
We also have the set G(Si), which includes all the accumulation
points of G(s). For sequences si in Si such that G(si) isn't
in R (e.g 1/x,1/x,1/x ....) we can reasonably define G(si) as
being _error_ For arbitary sets of functions G(Si) needn't
be uncountable, and can even be finite nor does G(S) have to be
a subset of G(Si).
We don't know much about G(S) or G(Si) for the particular set
Ca of calculator functions, and less for arbitary sets of functions.
Even for Ca we don't know if G(Si) includes Q.
The obvious way to prove G(Si) uncountable is to look at
how many elements of Si map to each element of G(Si).
If only a countable number do so then we know G(Si)
can't be countable else Si would be a countable union
of countable sets, which is not possible.
Because a subset of Ca maps the positive reals to themeselves
by considering this subset, which produces a subset of G(Si)
which doesn't include _error_ , we can place a lower bound on
the cardinality of G(Si) without having to consider _error_.
If the restricted subset can be proved uncountable then G(Si)
is also uncountable.
>
> 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_ }
>
We could also consider R+U{_error_}, C+ U{_error_}
or more abstractly mappings on arbitary fields.
Just considering R U {_error_} is challenging enough though.
Is anyone but Alberto reading this?
--
Robert
