Re: infinity

snapdragon31 wrote:
> Dave Seaman wrote:
> > On 7 Aug 2005 19:49:43 -0700, snapdragon31 wrote:
> >
> > > Dave Seaman wrote:
> > >> > There is no such a time that the vase would be empty.
> >
> > >> Wrong. Noon is such a time. If you disagree, name a ball that is in the
> > >> vase at noon.
> >
> > >> > I hope you can see that the argument "All balls are removed before
> > >> > noon." alone is not sufficient to claim that the vase is empty at noon.
> >
> > >> You keep asserting this, but you have never given any hint as to what it
> > >> is that makes you think such a thing. What is the reasoning behind your
> > >> outlandish claim?
> >
> > > It would be easier for you to see if the operation add only 1 ball (not
> > > 10) and then take out 1 ball. Assuming that we have 1 ball in the vase
> > > initially. You can see that no matter how many times there is always
> > > one ball in the vase. The number on the ball in the vase always
> > > changes.
> >
> > No, you have merely shown that there is always a ball in the vase after
> > each operation. But since there is no operation taking place at noon,
> > your argument is not sufficient to show that there is a ball in the vase
> > at noon.
> >
> > > At 11:59 Ball 1 is in the vase
> > > At 11:59:30 Ball 2 is added and ball 1 is removed. Ball 2 remains in
> > > the vase.
> > > At 11:59:45 Ball 3 is added and ball 2 is removed. Ball 3 leaves in
> > > the vase.
> >
> > > All ball comes into the vase will eventually moved out from the vase
> > > but at no time the vase is empty. Therefore, the vase can never be
> > > empty. Are you telling me after an infinite steps the vase will become
> > > empty? How? My suggestion would be after an infinite number of steps,
> > > there would still be 1 ball left in the vase. Can you give me a
> > > counter example?
> >
> > Again, when you say "at no time", you actually mean "at no time before
> > noon." In fact, each ball is removed before noon and therefore the vase
> > is empty at noon.
> >
> > Yes, I am telling you that the vase becomes empty at noon, because B(0) =
> > sum_{n=1^oo} B_n(0) = 0. Do the math.
> >
> > >> You also keep ignoring my question. If you think the vase is not empty
> > >> at noon, then what is the number of a ball that remains in the vase?
> >
> > > Not be able to name the number of a ball remains in the vase does not
> > > mean that the vase is empty. You can keep all the balls and only take
> > > away whatever ball I name at the very last step. It does not prove
> > > anything, right? I can prove that the number of balls grow
> > > indefinitely, but you cannot prove that the vase is empty.
> >
> > >> As long as you keep ignoring my question and refuse to provide any
> > >> supporting argument of your own, we will never make any progress.
> >
> > You still have not answered my question, and you still have not provided
> > a supporting argument of your own. I rest my case.
> >
> > --
> > Dave Seaman
> > Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
> > <http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
>
> Hi Dave,
>
> Please don't do the dirty trick again. Your question is to ask for the
> number of a ball remaining in the vase. My answer is that I cannot
> name the number of that ball. How come you always ignore my answer and
> claim that I ignore your question. I gave you the reason why. You say
> that I did not provide you a supporting argument. I don't mind if you
> don't agree with my point of view. This is the art (or the fun) of
> debating. If I can convince you or if you can convince me then this is
> not a good paradox. I have no intention to force you to take my point
> of view.
>
> >From my last reply:
> "Not be able to name the number of a ball remains in the vase does not
> mean that the vase is empty.

Maybe not, but we have something stronger here. For every ball n, we
know that ball n is not is the vase. This does mean the vase is
empty

> You can keep all the balls and only take
> away whatever ball I name at the very last step.

There is no very last step, so this is meaningless.

> It does not prove
> anything, right? I can prove that the number of balls grow
> indefinitely, but you cannot prove that the vase is empty."
>

The proof that the vase is empty is: "For every ball n, ball n is not
in the vase at noon. Therefore the vase is empty at noon.
The fact that the number of balls grows indefinitely (at finite steps)
has not been disputed by anyone. The fact that for every finite
step there are balls in the vase, but after an infinite number of steps
there are no balls in the vase is counterintuitive, but is is a fact.


> > Yes, I am telling you that the vase becomes empty at noon, because B(0) =
> > sum_{n=1^oo} B_n(0) = 0. Do the math.
>
> Please define what your symbols represent before I can do the math.
> My guess would be:
> B(0) - Number of balls in the vase at noon.
> sum_(n=1^oo) - Your operation A, balls added to the vase before noon
> B_n(0) - Your operation B, balls removed from the vase before noon
>
> If my guess is right, should there be a minus sign between the two
> operations? That is
> B(0) = sum_{n=1^oo} - B_n(0) = 0
>
> I gave a similar answer to someone. The detail is as follow
> Let S1 be the set of balls added to the vase
> S2 be the set of balls removed from the vase
>
> S1 = {1, 2, 3, ... , OO}
> S2 = {1, 2, 3, ... , OO}
>
> Balls remains in the vase would be
> S1 - S2

At this point we agree. And since S1-S2 is the empty set, the vase
is empty.

The problem is you take something like B a subset of A, C=A-B
and say "the number of elements of C is the number of elements of A
minus the number of elements of B" But this only works for
finite sets. For infinite sets this does not work.
So the statment

> S1 - S2= OO - OO the result is undefined and is not necessary equal to
> 0.
>

just means that we cannot determine the number of elements in S1-S2 by
using the number of elements in the sets. So we have to determine
S1 and S2 explicitely. When we do it turns out that S2 equals S1
(not only does it have the same number of elements, it has
the same elements).


> The result can be anything. If you want the result to be 0 then you
> need some kind of proof other than just saying it is.
>

Indeed. You need to prove that not only does S2 have the same number
of elements as S1 it has the same elements.


> For example if in each step only 1 ball is added and 10 balls are
> removed then the vase will be empty some time before noon provided that
> there are not infinite number of ball in the vase initially. It can be
> proved mathematically.
>

You start with an empty vase, add 1 ball and take away 10?

> If in each step 10 balls are removed first and then 1 ball is added
> back then there is no guarantee that the vase is empty at noon because
> after each step, exactly 1 ball will stay in the vase.
>

You start with an empty vase and remove 10 balls?

Let's go back to case A (start with all the balls, remove ball n
at step n).

The set of balls left in the vase is S1-S2, the set of balls
originally in or added to the vase minus the set of balls
removed from the vase.


At any finite step S1 is infinite and S2 is finite,
so there are an infinite number of balls in the vase.

After an infinite number of steps, S1 is infinite and S2 is
also infinite and equal to S1, so there are no balls in the vase.

-William Hughes

.

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