Re: Motion P1788.1/M004.01
On Thu, May 12, 2016 at 12:28 PM, Vincent Lefevre <vincent@xxxxxxxxxx> wrote:
> On 2016-05-10 13:17:40 -0400, Lee Winter wrote:
>> On Tue, May 10, 2016 at 10:46 AM, Ulrich Kulisch <ulrich.kulisch@xxxxxxx> wrote:
>>
>> > I think interval arithmetic should not be defined over the IEEE 754 binary64
>> > numbers. This more or less pulls all the IEEE 754 exceptions into interval
>> > arithmetic. We shoud not bother all users of interval arithmetic with
>> > constructs which really do not occur and definitly are not needed in
>> > interval arithmetic.
>> >
>> > Interval arithmetic shoud just be defined as a calculus for connected sets
>> > of real numbers.
>>
>> I disagree. We don't have native real numbers. We only have finite
>> (small) sets of rational numbers.
>
> I don't think that Ulrich meant *all* connected sets of real numbers.
The cardinality of the sets is a separate, and probably lesser, issue
from the nature of the elements of the set.
>
>> > They just serve as bounds for the description
>> > of unbounded real intervals.
>>
>> No. There are no unbounded real intervals. Every single computable
>> interval over rational FP numbers is bounded. We may lose track of
>> the bound through representation limitations, but there is such a
>> bound even if it has an unknown, or possibly unknowable, value.
>
> There are several possible definitions for "unbounded", e.g. the one
> from the order theory or the one for metric spaces (which also depends
> on the chosen metric).
As I understand them, neither applies to intervals for computer
software or computer hardware.
Consider an alternative description of unbounded in the sense that
there are members of the set that are not enumerable (in the NLC
sense).
Lee Winter
Nashua, New Hampshire
United States of America (RIP)