On "infinite" sets and accumulation points.
Yes, all actual computer representations of sets of numbers
are finite, but...
Vincent wrote, concerning "smallest superset" unambiguity:
> I think it is sufficient to assume that there are no
> accumulation points.
Dan Zuras wrote, replying to Vincent Lefèvre:
> > * Does IDBar have to be a finite set? ...
>
> First, no, it does not have to be a finite set. It is
> just finite in any implementation known or considered
There is still a difference between bounded and unbounded
formats in computers. In his context, "unbounded" actually
means something different from plain mathematics: it means
that the only bounds are resource constraints. And that is
significantly different from format-based constraints, and
deserves to be treated as if the sets really were "infinite".
The reason is that one wants to avoid runaway resource
consumption. Consider a program designed to compute the
smallest strictly positive real number. There are formats
where even exponents are represented in bignum form. Should
the program be allowed to consume all available resources, or
would it be better to detect early that there is no answer?
Note that ZERO is an accumulation point in many such systems.
So are many exactly-representable non-zero numbers.
Level-index systems are another case in point.
To get to P1788 more directly, this suggests that, if the
format permits "unbounded" precision, here should be controls
on actual precision in certain cases, e.g. "tightest hull".
(Alternatively one could indeed rule out such formats.)
Michel.
---Sent: 2010-07-13 14:59:15 UTC