Re: infinity

stephen@xxxxxxxxxx said:
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> > stephen@xxxxxxxxxx said:
> >> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> >> > stephen@xxxxxxxxxx said:
> >> >> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
> >> >> > Randy Poe said:
> >> >> >>
> >> >> >> Tony Orlow (aeo6) wrote:
> >> >> >> > because you claim there were balls in the vase, and then it became empty, and
> >> >> >> > you are only removing 1 ball at a time, so there MUST have been a last ball
> >> >> >> > removed. However, you cannot name that last ball removed, since there is always
> >> >> >> > another to remove. If you DID name that last ball removed, the I can name 9
> >> >> >> > times as many as have been removed that have not been removed. Don't you feel
> >> >> >> > like you are trying so sail a big sponge across the sea? Do you wonder why it's
> >> >> >> > such a drag?
> >> >> >>
> >> >> >> Can you name the last point between 0 and 1 which is less than 1?
> >> >> >>
> >> >> >> - Randy
> >> >> >>
> >> >> >>
> >> >> > Binary 0.111...111
> >> >> > Decimal 0.999...999
> >> >>
> >> >> What is (0.999...999+1)/2? Is it between 0.999...999 and 1?
> >> >> Or is it undefined?
> >> >>
> >> >> Stephen
> >> >>
> >> > 000...000:000...000.999...999:500...000. Very good. You got half an
> >> > infinitesimal out of me! ;D
> >> > --
> >> > Smiles,
> >>
> >> > Tony
> >>
> >> Is it between 0.999...999 and 1? If so, then 0.999...999 is
> >> not the last point between 0 and 1 which is less than 1.
> > 0.999...999 is the last first-level infinitesimal in the first unit. You have
> > just dipped your toe into the second level of infinitesimals.
>
> The question was about 'points'. Are second level infinitesimals
> points or not? They apparently are located on your number
> line between other points.
Second level infinitesimals are to points what points are to lines. They are -1
dimensional entities, and infinity of which make up a point.
>
> >>
> >> If it is not between 0.999.999 and 1, then where is it?
> > That's where it is, in between those two points which you consider identical.
>
> So were you wrong when you claimed that .999...999 is the
> last point between 0 and 1 that is less than 1?

It is the last whole point.

> There
> are apparently an infinite number of second level infinitesimals
> between .999...999 and 1. And I imagine there are even
> more third level, fourth level, etc level infinitesimals
> lurking in your number line. Are all of these infinitesimals
> points or not? If they are not points, what are they?
negative dimensional objects.
>
> >> You are going to have to redo a lot of basic arithmetic
> >> as x<y -> x < (x+y)/2 < y is a direct consequence of basic
> >> arithmetic.
> > That still holds.
>
> Only because .999...999 is not the last point between 0 and
> 1 that is less than 1. I wonder if you are now able
> to recognize that there cannot be a last point between 0 and
> 1 that is less than 1, unless you give up closure under
> addition and division.
It depends what levels you are allowing, it seems.
>
> Stephen
>

--
Smiles,

Tony
.

Relevant Pages

  • Re: infinity
    ... > Tony Orlow wrote: ... >> Second level infinitesimals are to points what points are to lines. ... > Yet you've stated previously that points have no (zero) width. ...
    (sci.math)
  • Re: infinity
    ... > Second level infinitesimals are to points what points are to lines. ... Yet you've stated previously that points have no (zero) width. ... kind, not the infinitesimal kind)? ...
    (sci.math)
  • Re: infinity
    ... > 0.999...999 is the last first-level infinitesimal in the first unit. ... > just dipped your toe into the second level of infinitesimals. ...
    (sci.math)