Re: P1788: Our first formal motion has entered its discussion period

[Ulrich Kulisch <Ulrich.Kulisch@xxxxxxxxxxx>]

Arnold Neumaier schrieb:
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
>
> 1. Ulrich Kulisch wrote:
> > Concerning StandardizedIntervalNotation I would like to make two 
> > remarks:
> > (in this mail R stand for the set of real numbers and I for an 
> > interval set.)
> >
> > 1. Of course basic to all considerations is the set of extended reals 
> > R*.
> > Interval arithmetic deals with closed and connected sets of real 
> > numbers (and nothing else). If an interval is bounded it is written 
> > as [a, b] with a, b elements of R. If it is unbounded it is written 
> > as (-oo, a] or [b, +oo) with a, b elements of R or (-oo, +oo) where 
> > the parantheses indicate that the bounds -oo and +oo are not elements 
> > of the interval. The set of all such intervals should be denoted by 
> > IR. Then
> > {IR, +, -, *, /} is an exception free calculus.
> >
> > So what do the proposed notations like IR* or *IR^n really mean? If 
> > we really consider intervals of IR* we would have to allow intervals 
> > like [-oo,-oo] and [+oo, +oo]. 
>
> These are in IR^* but not in IR. Standard interval arithmetic is for IR;
> so there is no conflict here.
>
>
> Arnold Neumaier
I cite from the proposed StandardNotation:

A (real, closed, nonempty) interval is a 1-dimensional box, i.e., a pair x =
[x, x] consisting of two real numbers x and x with x ? x. The set of all
intervals is denoted by IR.

(I am sorry. The underlines and overlines have not been copied in the 
cited text.)

In my understanding this definition excludes the empty set and intervals 
like (-oo, a] or [b, +oo) with a, b elements of R or (-oo, +oo) from IR. 
In my definition above these are included in IR.

Ulrich Kulisch