Re: Disjoint, subset, & interior, or more...???

[Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>]

Nate Hayes wrote:
> George Corliss wrote:
> > I agree.  I suspect one factor contributing to the relatively low 
> > voter turnout on 13 & 21 (did I say PLEASE VOTE) may be information 
> > overload. Let's let a few of these run to completion.
>
> I agree, too.
>
> I also have the observation that yet another problem contributing to 
> confusion is that comparision relations are affected by wether intervals 
> such as [1,Infinity] are unbounded or bounded (say, because of an 
> IsBounded decoration or Overflow, etc.).

According to Motion 3, and in accordance with standard mathematical
practice, [1,Inf] is unbounded, no matter how decorations handle this.

If we cannot rely on accepted motions to discuss further motion
we'll always remain on ground zero and never get anywhere.


> For example, on the one hand Arnold argues that the "interior" relation 
> in Motion 13.04 is not topological interior. THis criticism is valid 
> only for unbounded intervals. On the other hand, Arnold also advocates 
> an IsBounded deocration. In that case, the definition for "interior" in 
> Motion 13.04 _is_ the correct definition of topological interior (by his 
> own logic and reasoning as shown in recent e-mails in this forum).

No. No matter how intervals are represented, the inequality x>=1 always
defines the unbounded interval [1,Inf], and not a bounded surrogate
[1,Overflow] without a meaning as a set of real numbers.

And it is _this_ interpretation that is needed in the applications in
optimization.

To be able to apply mathematical theorems to such intervals we need
their mathematical meaning, not an ad hoc meaning created by the wish
to do some computations more efficiently. If the meaning of isInteriorTo
does not match its mathematical meaning in case of an interval [1,Inf]
defined by the constraint x>=1, its implementation is worthless.


> So some of these confusing aspects I suspect are not going to go away 
> until P1788 tackles these other related issues.

Motion 3 already settled the meaning of [1,Inf].

Confusion appears only if accepted motions are ignored in the
discussion.