Re: infinity


Kirby Cook wrote:
> William Hughes wrote:
>
> > Kirby Cook wrote:
> >
> >
> >> Your challenge is persuasive; I cannot answer it. Can you answer when
> >>I ask, by what logic you make 9 + 9 + 9 +... =0?
> >>
> >
> >
> > You seem to have pulled the infinite series
> > 9 + 9 + 9 + ... out of midair.
>
> Oh, come on! At the end of each step, beginning with the first one,
> there are nine more balls in the vase that there were at the end of the
> preceding step.

Indeed and no one has disputed this. However, this is it self does
not lead to an infinite series and no one but you has introduced an
infinite series.

>Is there indeed something wrong with writing
> (10-1)+(10-1)=9+9? Is there a problem with the ellipsis?
>

Only that this obsures the fact that the series

10 - 1 + 10 - 1 ...

is only conditionally divergent to infinity. I could rearange an
infinite number of terms to get

( 10-1-1-1-1-1-1-1-1-1-1) + ( 10-1-1-1-1-1-1-1-1-1-1) ...
= 0 + 0 + ... = 0

But all this is really beside the point. No infinite series is
a good model of what is happening.

> I assume you mean something like
> > a(n) is the number of balls added at step n
> > sum_1^k [a(n)] is the number of balls in the vase at step k.
> >
> > Some comments
> >
> > -the conditionally divergent series
> >
> > 10 - 1 + 10 -1 ...
> >
> > is probably a better model
> >
> > -in any case the sum of the series lim(k->inf) sum_1_k [a(n)]
> > (even if defined) is not the number of balls in the vase at
> > noon.
>
> Why not? Believe it or not, that's not a challenge, but a question.

I explain below. If we use as a criterion for emptyness that every
ball that is put into the vase is removed from the vase, then the
number of balls in the vase at any finite step is not relevant.
So the sum of an infinite series (which is determined by
the number of balls in the vase at finite steps) is not relevant.

> Seaman mentioned a function discontinuous at zero. What does that mean?

Merely that, in this problem, the function representing the number
of balls in the vase is discontinuous at noon. Note that Dave Seaman
defines

B_n(t) = 1, if ball n is in the vase at t,
0 if ball n is not in the vase at t.

(Note in this problem, each ball in put in the vase once, and removed
at most once. So for each n,
B_n(t) is simple function with either one or two steps)

Dave Seaman further defines the number of balls in the vase at any time
t to be

B(t) = sum_1_infinty B_n(t)

At this point we are basically done. B(t) has been defined, the
only thing the remains is to calculate it. When you do you find
that B(t) is discontinous at noon. If you do not like the result,
quarrel with the definitions of B_n(t) and B(t).

> Or rather, if for a given function of x, f(x) approaches infinity as x
> approaches a given value, the function is said to be discontinuous at
> that value, right? Which means, at that value, the function doesn't
> apply? Doesn't exist? What? It seems to me that, at that value,
> speaking of any result in terms of that function would be meaningless.
> But when you say that there are zero balls in the vase at noon, that
> implies, to me, that there is some bridge between what went before and
> where you find yourself at noon and, indeed, you define your answer in
> terms of what went before, that is, that every ball added is due to be
> taken out at some later time. OK. I have trouble ignoring the
> concurrent condition that, at that later time,in every case, there are
> 90 more balls in the vase than were there when the ball was added.
>

You may have trouble ignoring the fact that the number of balls
in the vase increases without bound at the finite steps, but the
fact of the matter is that once you come up with a reasonable defintion
of the number of balls in the vase after an infinite number of
steps. it turns out that this number does not depend on the
number of balls in the vase after a finite number of steps.

> >
> > We need to know how many balls are in the vase after an
> > infinite number of steps. We say for ball n:
> >
> > -if there is a step u(n), such that ball n is removed
> > from the vase at this step and not subsequently added
> > ball n is not in the vase after an infinite number of steps
> >
> > -if there is a step v(n) sub that ball n is added to the
> > vase at this step and not subsequently removed, ball n
> > is in the vase after an infinite nubmer of steps
> >
> > -if neither of the above is true, we say that it is
> > not defined as to whether ball n is in the vase or
> > not after an infinite number of steps.
> >
> > Note, the above does not refer to the number of balls in the vase
> > after a finite number of steps. So the number of balls in the vase
> > after a finite number of steps DOES NOT MATTER. Nor does any limit
> > involving the number of balls in the vase after a finite number of
> > steps. So saying that the vase is empty after an infinite number
> > of steps does not say anything about an inifinite sum.
> >
> > -William Hughes
> >
> William, The problem here, as I see it, is that "an infinite number of
> steps" is undefined, or inadaquately defined.
>

"an infinite number of steps" is defined to be one step for each
positive integer. It is as well or poorly defined as is the set
of positive integers. You can claim that the set of positive integers
is not well defined, but do not expect me to agree with you.

-William Hughes

.

Relevant Pages

  • Re: infinity
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  • Re: infinity
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  • Re: infinity
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