Re: Motion P1788/M0032.01:MidpointMeaning -- discussion period begins
> I would accept a change that it is limited to
> bounded non-empty intervals at level 1.
Thank you. I agree entirely with the modified motion.
-Dima
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Отправленные: Суббота, 10 Март 2012 г 1:01:56 GMT +04:00 Абу-Даби, Маскат
Тема: Re: Motion P1788/M0032.01:MidpointMeaning -- discussion period begins
> Date: Fri, 9 Mar 2012 05:47:16 -0800 (PST)
> From: Dmitry Nadezhin <dmitry.nadezhin@xxxxxxxxxx>
> To: <intervals08@xxxxxxxxxxxxxx>
> Cc: <stds-1788@xxxxxxxxxxxxxxxxx>
> Subject: Re: Motion P1788/M0032.01:MidpointMeaning -- discussion period begins
>
> Dan,
>
> I agree with your Level 2 definition of mid .
>
> And I have a small note on level 1 definition.
> John's Level 1 draft defines domain level 1 "mid" function as
> nonempty bounded intervals only.
> This motion defines it for semi-infinite intervals also.
This is so & I was hoping no one would notice. :-)
>
> This means that Level 1 mid(X)=mid([u,+oo[)=(u + oo)/2 = +oo .
> This seems natural.
> However, it violates the formulated property
> mid(X) \element-of (X1 \intersect X2)
> +oo can't belong to (X1 \intersect X2) because level 1 intervals are
> sets of reals.
>
> Is the domain of Level 1 definition of "mid" in the scope of this motion ?
It is & I would accept a change that it is limited to
bounded non-empty intervals at level 1.
>
> -Dima
>
Preamble to all: In order to address Dmitry's comment
on its merits, I am about to discuss some controversial
stuff that has absolutely NOTHING to do with the motion
in question. Please do not start a long discussion in
this forum. If you must, talk amongst yourselves.
Dmitry,
It DOES violate the notion of semi-infinite sets that
live at level 1. At least, those that live within IR.
And if you would like me to modify the proposal to
include only bounded non-empty intervals within IR at
level 1, I would accept that as friendly. They can
be reintroduced at level 2 in much the same way as
midpoint(Entire) was done.
Frankly I did it intentionally hoping no one would
notice. You see, I have been struggling with issues
about level 1 that are in the relm of analysis & limits
& other things about how we treat infinity. I have
never wanted to discuss them because I feel they are
largely irrelevant to our standards work & I did not
want to start yet another lengthy philosophical
argument with no good resolution or utility to our
work.
Still, now that you have asked, level 1 also has
intervals that live in IRbar. And, strictly speaking,
that would mean sets that include infinity as an element.
But we have chosen to take the interpretation that a
semi-infinite interval is open at the far end such that
infinity is NOT an element of that set. Fine & good.
I can live with that.
However, we still have some funny sets that live at
level 1. Within IR we have sets like bigun(r) = [r,2r].
That means, for any finite set, X, you can name, there
is an s > sup(X) such that bigun(r) > X for all r > s.
Then within IRbar, the set bigend = limit (r-->inf) bigun(r)
can be defined. It is a set that contains no Real
elements yet it is different than the empty set in that
it is known to be larger than any finite set you can
name.
A funny set. At least to me.
One could also consider the implications of semi-infinite
sets like bigsemi(r) = [r,+inf) with slightly different
lines of reasoning.
Anyway, rather than introduce these notions to the
pointless controversy I KNOW they would engender, I
fudged on the definition.
I can go back if you think its important.
For Juergen & Baker, the changes would be to limit the
scope of the level 1 midpoint to bounded non-empty
intervals. And then to introduce the semi-infinites &
Entire at level 2 as projections onto some finite
explicit interval system IF over some floating-point
system F.
But we need NOT have a big argument about it.
Please.
Dan