3rd Experiment |OR| Infinity
Joubin Houshyar
My friend, who doesn't like sticking his hand in a vase (some sort of
phobia, I think he mumbled something about cantorophobia ...) and I
performed the following experiment:
We obtained two Giant Vases.
We obtained two Big Bags of Balls.
We obtained two Magic Markers.
Each of us used one of the Giant Vases, and one of Big Bag of Balls,
and one of the Magic Markers.
It was near noon when we both grabed 10 balls from our respective Big
Bag of Balls, and were about to write the numbers 1 through 10 on them
to get started for the experiment. Just as we were to perform the
process as outlined by the original poster my friend sheepishly
admitted to his phobia and suggested what sounded like an innocent
alternative.
He said:
"You go ahead, and FIRST Mark the balls '1' through '10' and then put
them in the vase and then remove the ball Marked '1'. I'll simply grab
10 balls and put in 9 balls in my Big Vase and Mark the remaining ball
as '1'. And so on. It should be the same, no?"
Now he may be a bit of a frighty rabbit when it comes to sticking his
hands in Big Vases, but he can be a tricky sort of guy. So I wasn't
sure if we were both actually doing the 'same' thing. But, he insisted
that our actions, at each step, were equivelant. To prove it, he
suggested we try it out for a Finite set of steps, and sure enough,
after the finite set of steps were taken, we both had in front of us
balls '1', '2', '3', ..., 'n'.
So that convinced me and I said, ok, now lets do the experiment as
outlined.
At one minute to noon, we then proceeded as he had suggested ...
When noon came around, we both had a whole bunch of balls in front of
us.
You Name the 'n', and there it was, right there and obviously NOT in
(respective) Big Vase. (This was true whether you looked at his set or
mine.)
I said, OK, that settles it. My Big Vase must be empty since you can't
'Name' a single 'n' that is not in front of me.
He went to his Big Vase and started shaking it, and sure enough, there
were balls in there. (After all, he never removed anything from his
Big Vase.)
Then he told me to shake my Big Vase. (And that's when I sheepishly
admitted to _my_ phobia of shaking Big Vases ...)
What's going on? My Big Vase is empty, Right? But how could that be,
since we were doing the equivelant action in each and every step?
/R
Joubin Houshyar
-- begin original --
Hello everyone,
I'm having an argument with a friend about the following problem:
Suppose you have a giant vase and a bunch of ping pong balls with an
integer written on each one, e.g. just like the lottery, so the balls
are numbered 1, 2, 3, ... and so on. At one minute to noon you put
balls 1 to 10 in the vase and take out number 1. At half a minute to
noon you put balls 11 - 20 in the vase and take out number 2. At one
quarter minute to noon you put balls 21 - 30 in the vase and take out
number 3. Continue in this fashion. Obviously this is physically
impossible, but you get the idea. Now the question is this: At noon,
how many ping pong balls are in the vase?
An 2nd experiment goes as follows:
put no. 1 - 10 in, take no. 10 out, put no. 11-20 in, take no. 20 out,
put
no. 21-30 in, take no. 30 out, etc.
(of course at the same moments as above)
My friend claims both experiments end up with an empty vase.
I think however the 2nd experiment ends up with a vase with an infinite
number of balls:
1-9, 11-19, 21-29 etc. are definitely in the vase.
He says it all has to do with Cantor's set theory, cardinality etc...,
but
browsing the internet didn't really help me much.
Any information or relevant links are very welcome,
Thanks, Theo
-- end --
Ross A. Finlayson
Frickin' jokers.
Hmm... you got infinitely many balls. Basically, the factory serial
number on his set is in base 19 and on your set base 10, and it so
happens that the numbers match those you draw in a happenstance way.
So your friend's ball 100 is actually 19^2 or something, not 10^2, of
the matched set, and a bunch of his escaped the marker.
How about this, at each step put n two and take one back. At noon,
dump em out there's twice as many. Oh, wait, you put in half so there
are just as many as there were, x/x = 1.
Maybe you should go drinking and _then_ think about it. Try to not
disturb too many street signs in moderation.
In the second experiment, you have one tenth of the balls. If you dump
them back in and randomly select one from the vase, there's a one in
ten chance that it's one of those you had set aside, because the
asymptotic density of the multiples of ten or counting integers
congruent to 0-9 mod ten is one tenth.
In the first experiment, that's kind of undefined. Basically it would
have to be some form of "nonstandard" density, and it's not totally
obvious how to compare that to other "nonstandard" densities. It seems
that you have to dump them back into the vase and then select a random
ball. It's kind of like, at each step you dump out all the balls, and
then put back in all the balls at that point, instead of being able to
select one when you are in infinite speed mode. When you're moving
that fast the vase is opaque. You can shake the vase and select one,
or push a button and they all dump out, but to just get the one you
want you have to keep picking them one, at a time, at random until you
get the one you want. You can dump in as many as you want at a time,
and look at the ones not in the vase, but you can't look in the vase,
and can only get one at a time randomly or dump the entire vase.
This is where every set is measurable because a well-ordering of the
reals implies a least positive real, and their sum over the natural
integers is two, instead of one, because they were considered one at a
time instead of all at once, on a two-dimensional line as opposed to,
say, the unit disc.
Good day,
Ross
Timothy Little
> "You go ahead, and FIRST Mark the balls '1' through '10' and then put
> them in the vase and then remove the ball Marked '1'. I'll simply grab
> 10 balls and put in 9 balls in my Big Vase and Mark the remaining ball
> as '1'. And so on. It should be the same, no?"
> What's going on? My Big Vase is empty, Right? But how could that be,
> since we were doing the equivelant action in each and every step?
At any time before noon, they're equivalent (in cardinality). At
noon, they're not. Although it is possible to label all the balls at
the end of any single step so that the situation is identical, it is
not possible to do so in a way that carries over to future steps. The
limit is defined in terms of tracking the movements of any given ball
(where such a limit exists)*.
So in short, you're not really doing fully equivalent actions at each
and every step, in terms of the definition of the limit.
* An example where the limit doesn't exist: at step 2n-1, put in ball
1. At step 2n, take out ball 1. At noon, there is nothing to say
whether ball 1 is in the vase or not.
Similar example where the limit does exist: at step 2n-1, put in ball
n. At step 2n, take out ball n. At noon, you can confidently say
that every ball is not in the vase.
- Tim
pfn...@yahoo.com
At every step you are adding 10 balls and removing 1. This is the same
as saying that you are adding 9 balls at every step. If you repeat this
ad infinitum you end up with an infinite amount of balls in the vase.
Simple as pie.
The corresponding mathematical expression to find the number of balls
after the nth step is
10*n - 1*n
which is obviously the same as 9*n. To find how many balls are there in
the vase at infinity (or as we approach infinity, as you prefer) we
take the following limit:
lim n->oo (10*n-1*n)
Well, if you don't simplify 10*n-1*n, you get with an expression oo-oo
which as we know is *indeterminate*, and which I believe to be the
hidden cause of the debate.
In the other hand, if we simplify the limit to
limit n->oo 9*n
the answer should be obvious.
Oh, and please carry out the experiment with transparent vases. That
should help make evident that we are not getting any closer to
emptiness the more steps we take, and it should help with your friend's
phobia too (full reference:
http://groups.google.com/group/sci.math/browse_frm/thread/2ffb3c24245d7803/a80c935e2d7b1c40#a80c935e2d7b1c40
).
Cheers,
--Tech
Joubin Houshyar
You are analyzing the original experiments #1 and #2.
The problem at hand is the 3rd experiment as outlined.
I claim my Giant Vase will be found empty on, or, at any time after,
noon. And I have 100 years of unassailable mathematics to "prove" it.
To anyone who claims my Giant Vase contains even a single ball, I
challange you to Name that 'n'! For any 'n' that you can 'Name', I can
pick it out from the pile in front of me.
He has 'demonstrated' that his Giant Vase is NOT empty. And of course
it isn't 'Magic'! He never removed anything from his Giant Vase. Now
what he is claiming is that *my* Giant Vase is not empty and is being
very (very) aggressive about wanting to shake my Giant Vase (which I
refuse on 'Principle'!)
So this is where we are stuck:
If he challanges me to shake my Giant Vase (to prove there are balls in
there) I tell him, "Nonsense! 'Name' the 'n' that you propose is in my
Giant Vase!". And he can't.
If *I* challange him that we did NOT perform the same task afterall,
*he* challanges me "Oh yeah? Then 'Name' the 'n' that you claim is in
your pile in front of you that is NOT in my pile!" And this is where I
am stuck: FOR ANY 'n' that I can 'Name', it is found BOTH in his pile
and my pile!
We both agree that we did NOT perform the 'exact' 'same' operations,
but that they are equivelant.
I've tried to use the *fact* that he does NOT 'Mark' every ball whereas
I do -- obviously NONE of the balls in his Giant Vase are 'Marked' --
but then he smirks and says "Mysticism doesn't suite you. Lets just
stick to math, shall we?"
I've also tried to use the *fact* that his 'partitioning' of the
unit-set -- you know, the 10 un-Marked balls we take out of the Giant
Bag of Balls -- at each step occurs 'Outside' of the Giant Vase (given
his alleged 'phobia' (which now I suspect is just a ruse)), whereas I
do this 'Inside' the Giant Vase. But when I point that out to him, he
starts shaking his head and says "Oh dear! Are we doing Math or
Meta-Physics?"
Finally, I grab at the *last* difference in the situation and tell him
"Ok, since the balls inside of your Giant Vase are not 'Marked', how
would you 'Name' them uniquely given that the pile in front of you is
N?" And that is when he starts laughing like a mad man and says
"HAHAHAH ... Oooh I could 'Name' them, but you wouldn't like it!"
So there we are:
We have taken 'similar' but not identical actions. For any
(arbitrarily large) number of steps 'n', it can be demonstrated that
both our Giant Vases contains the *exact* same number of balls, and, we
both have an *identical* pile of balls marked '1', '2', ..., 'n'.
Then, AFTER the expermient is finished, we have:
2 Giant Vases: one 'demonstrated' to be apparently full of un-Marked
balls; one 'asserted' to be empty.
2 Giant Bags of Balls: both now completely empty.
2 big pile of 'Named' balls: These appear to be 'identical'. (At
least no one has been able to Name an 'n' that is in my pile that is
also NOT in his pile. Further no one (including myself) can come up
with any function that would produce a 'n' that is in my pile but NOT
in his. (Try it.))
So, this guy wears a shirt with a picture of Poincare on the front and
a cryptic "Wot's in 'Name'?" on the back ... just to give you an idea
what I'm dealing with here ... and he has written the following letter
[1] on his Giant Vase. He points to his Giant Vase and says "There is
my Tauf-Nut!" and then start to laugh!
Its very annoying.
/R
Joubin Houshyar
Shmuel (Seymour J.) Metz
10/04/2005
at 06:09 AM, pfn...@yahoo.com said:
>Obviously this is physically impossible, but you get the idea.
No; you have defined neither an experiment nor a mathematical
question.
>which is obviously the same as 9*n. To find how many balls are there
>in the vase at infinity
What does that mean? You have defined neither a physical experiment
nor a mathematical question.
>If you repeat this ad infinitum
What does that mean? You have defined neither a physical experiment
nor a mathematical question.
>To find how many balls are there in
>the vase at infinity (or as we approach infinity, as you prefer)
First you have to define what you mean by that.
>we take the following limit:
That will tell you what the limit is; it won't answer unrelated
questions.
>Well, if you don't simplify 10*n-1*n, you get with an expression
>oo-oo
No you don't.
>and which I believe to be the hidden cause of the debate.
The cause the debate is not hidden. The cause of the debate is the
refusal to use precise language and to consistently use the same
definitions for words throughout the argument. The cardinality of a
set has nothing to do with the limit of an unrelated sequence.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
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