Re: An uncountable countable set

Tony Orlow wrote:
Virgil wrote:
In article <45215d2f@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

Han de Bruijn wrote:
stephen@xxxxxxxxxx wrote:

how many balls are in the vase at noon?

What does your "mathematics" say the answer to this
question is, in the "limit" as n approaches infinity?
My mathematics says that it is an ill-posed question. And it doesn't
give an answer to ill-posed questions.

Han de Bruijn

Actually, that question is not ill-posed, and has a clear answer. The
vase will be empty, if there is any limit on the number of balls, and
balls can be removed before more balls are added, but it is not the
original problem, which states clearly that ten balls are inserted,
before each one that is removed. That's the salient property of the
gedanken. Any other scheme, such as labeling the balls and applying
transfinitology, violates this basic sequential property, and so is a ruse.

One can pose any gedanken one likes.

If TO does not like to be able to tell one ball from another, he does
not have to play the game, but he should not ever try to pull that in
games of pool or billiards.

If distinguishing balls gives a less exact answer,

Less exact how?

and a nonsensical one to boot

It makes sense to me that if you put a ball into a vase and later
remove it then it isn't there. It also makes sense to me that if you
put a ball in a vase and don't remove it then it is still there. What
*doesn't* make sense to me is that if you put some number of balls in a
vase and remove them all then there are still some left. That seems to
be what you are claiming.

Note : I agree with those who say it makes no sense in physical terms
to have an infinite number of balls. But mathematics is an idealisation
so it can make sense to talk about the infinite, even if it is
physically impossible.

then that attention can be judged to be ill spent, and not
contributing to a solution at all. It is clear that sum(x=1->oo: 9)
diverges, is infinite, not 0. It's ridiculous to think otherwise.

But the number of balls in the vase at noon *isn't* the limit of that
sum, Tony. Nobody disagrees that that sum diverges (of course, we might
disagree that it diverges to a "specific actually infinite value", but
I digress...), people disagree that the limit of that sum is the same
thing as the number of balls in the vase at noon.

It seems a little barmy to spend a year arguing something that nobody
disagrees with - that the sum 9, 18, 27... diverges. You should instead
try to argue that the limit of that sum is equal to the number of balls
in the vase at noon - that is what people are disagreeing with! Just
saying "clearly" doesn't quite cut it.

Here's my take on things...

Problem : at one minute to noon, balls 1 thru 10 are added to the vase
and ball 1 is removed. At half a minute to noon balls 11 thru 20 are
added and ball 2 is removed. etc.

Let noon = 0 and "one minute to noon" = -1.

Let A(n,t) be 1 if the ball n is in the vase at time t, 0 if it is not
in the vase at time t.

Let B(n) be the time that the nth ball is added to the vase and C(n) be
the time that it is removed.

B(n) = -1/(2^(floor((n-1)/10)))
C(n) = -1/(2^(n-1))

Note that B(n) and C(n) are strictly less than 0.

Now A(n,t) = { 1 if B(n) <= t < C(n)
0 otherwise }

Note that A(n,0) = 0.

Let S(t) be the number of balls in the vase at time t. Then

S(t) = { sum(n=1..) A(n,t) }

Then

S(0) = { sum (n=1..) A(n,0) }
= { sum (n=1..) 0 }
= 0

QED.

Now I'm going to look a little at your argument, based on limits.

Let's look at the sequence of S_ns where

S_n = S(-1/(2^(n-1))

That is, S_n is the number of balls in the vase "just after" the nth
iteration. Then, you say, the limit of the sequence of S_ns must equal
S(0) (ie, the number of balls at noon) and the limit of the sequences
of S_ns (9, 18, 27,...) is "positive infinity" and thus the number of
balls at noon is infinite. But this is fallacious reasoning, Tony! What
justification do you have for asserting that the limit of the S_ns is
equal to S(0)? Presumably something along the lines of...

+ oo = lim((n->oo), S_n)
= lim((n->oo), S(-1/(2^(n-1)))
= S(lim(n->oo), -1/(2^(n-1)))
= S(0)

But this is not a valid manipulation of limits. lim(n->oo,S(T(n))) =
S(lim(n->oo),T(n)) is true when S is continuous. But S is not
everywhere continuous in our problem. So you have no justification for
your assertion that S(0) is equal to the limit of the S_ns.

--
mike.

.

Relevant Pages

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