Seems reasonable that the shorter stick dwindles to zero, and the longer stick increases to unit length. I’m sure there’s got to be a convergent series that sums to one for the longer stick, and subtracts from the 1/3 towards 0 for the shorter stick.
You’re creating an infinite number of shorter and shorter length sticks, but you’re never tossing them away. They simply move from the shorter stick side to the longer stick side. After infinite time, you’ve got nothing on the one side, and a whole stick on the other.
Boy, I haven’t been paying attention this week, have I?
That’s not the point of my previous post. I agree that in your new setup, it is no longer possible to say with certainty that there are no balls in the urn “at time infinity”. What I’m arguing is that your new setup is not the same as the setup in the OP, because an observer inside urn A in your setup no longer has enough information to determine which balls are in urn A and which aren’t.
Which leads to your next point…
ultrafilter has already addressed this somewhat, but let me repeat the following: the answer to “how many” depends on the answer to “which”.
That’s why we keep bringing up the case of when the gremlin always removes the highest-numbered ball. No one’s arguing that in that case, there will be an infinite number of balls remaining. But that case is fundamentally different from the case where the lowest-numbered ball gets removed, precisely because the answer to “which” is different.
I assume this is a typo, since urn B only holds two balls.
The rest of that post seems to have already been addressed, so moving on to a later post…
No. I would argue that there is one stick and the total length is one. But this situation also differs from the OP.
In the OP, after each minute there was a collection of balls in the urn, and these balls came from another larger collection consisting of all balls that could possibly be in the urn. The balls themselves are presumably discrete, indivisible objects. Hence it is appropriate in this case to model the collection of all possible balls with a set of discrete elements such as the positive integers, and to consider the balls in the urn at any given time as a subset of this larger set.
But in the new situation you describe, the sticks aren’t discrete, indivisible objects, and the sticks held by the gremlin at time n aren’t drawn from some large collection of all possible sticks. Instead, in this case it is appropriate to model the original large stick with a continuous set with a notion of “length”, such as the interval from 0 to 1 in the real line. Then the two sticks the gremlin holds at any time can be considered as subsets of this larger set.
So in this new situation, there are two sequences of sets, one for each of the two sticks at any given time. The first sequence (using the standard notation [a,b] to denote the interval from a to b) is the sequence [0,1/3], [0,1/4], [0,1/5], …, which converges to the empty set. The second sequence is the sequence [1/3,1], [1/4,1], [1/5,1], …, which converges to the whole interval [0,1].
Which is why I would say that in the limit there is only one stick, and that that stick has length one.
I go away from this thread for a week and my faculties go to pot…let me correct myself.
The limit of the first sequence is not the empty set, it is the zero-length interval containing only the point 0. And the limit of the second sequence is not the whole interval [0,1], but instead is the set of all points in [0,1] except the point 0.
I would still say that in the limit there is only one stick and that that stick has length one, if only because I don’t think a stick of length zero really deserves to be considered a stick…
Every once in a while I like to stop lurking and stick my two cents in. It seems to me that the fundamental mistake being made by ZenBeam and others who think that there are infinitely many balls left in the urn is assuming that the number of balls “in the limit” is the limit of the number of balls at each stage. But cardinality is not “continuous” in this way. Consider a very simple sequence of sets {S[sub]n[/sub]} with S[sub]n[/sub] being the set of all integers n or larger. The limit of this sequence of sets under any reasonable definition will be their intersection, which is empty, so the number of elements in the limit is 0. However, each set in the sequence is infinite, so here’s a rather gross failure of the assumption that the cardinality of the limit is the limit of the cardinalities. This is why the question is not simply “how many” but “which”, since concentrating on “how many” can’t resolve the problem.
Any assumption that a property or measurement is preserved in the limit is automatically suspect and false until proven otherwise. This is the nature of limits and the source of a lot of interesting mathematics.
Speaking of measures, ultrafilter, I suspect you’re right about the probability of a finite number of balls being left being 0 in the random case, but your explanation is a little too blithe. It’s not true that countable sets always have measure 0; it depends on the measure. I would want to see an analysis first of what measure on the set of subsets of the integers we’re getting by this procedure. But it’s getting late and my brain isn’t clicking on enough cylinders to begin that analysis…
ultrafilter, because this may be relevant to analyzing the random case, let me point out a mistake in your proof that any set S may be realized by the goblin: The result is true only if, at each step, you allow the goblin the option of not removing any ball. When you say that the goblin removes the lowest-numbered ball not in S, there need not be any such ball to remove. If, instead, you insist that the goblin always be able to remove a ball, then the realizable sets S are those that, for each n, contain at most half of the numbers from 1 through 2n.
Now, for convenience let me give the name goblin measure to the measure on the set of subsets of N (the natural numbers) implied by the procedure of adding two balls, numbered 2n-1 and 2n, at stage n, and having the goblin remove one of the balls remaining in the urn randomly (with all remaining balls equally likely). I make one observation: The probability of any given ball remaining to the limit is 0 (exercise for the reader). Note that this does not imply that the limit set must be empty. However, on this flimsy evidence I make the following two conjectures:
[ul]
[li]Weak Goblin Conjecture The empty set has nonzero goblin measure. That is, there is a nonzero probability that the goblin will leave an empty set when all is said and done.[/li][li]Strong Goblin Conjecture The empty set has goblin measure 1.[/li][/ul]
I’m not at all sure I believe the Strong Goblin Conjecture, but I throw it out there because I’m feeling reckless.
The lengths of the sticks form a sequence of sets, S[sub]n[/sub], n = 1,2,…, each of which consists of two numbers: 1/(n+2) and 1 - 1/(n+2). For that sequence, you have already replied:
How do you reconcile those two statements?
Yes, that’s a typo. Let me try again: Even though which balls are in urn A is not defined, is anyone here willing to say there is not an infinite number of balls in urn A? I may as well also ask: Does everyone agree that urn B holds two balls, even though you can’t say which two?
Just because you can’t answer “which” doesn’t mean the answer to “how many” is zero.
Hmm…I thought I said that the gremlin would remove no ball if there was no ball not in S left in the urn. If I didn’t, I meant to say that.
Measure theory isn’t necessarily my strong suit, so I’ll defer to you on that matter. I had assumed that all measures were similar to the Lebesgue measure, but I had forgotten about, say, the counting measure. Ah well.
The set of stick lengths goes to the null set, but their sum is always one, and every point from [0, 1] is contained in exactly one of the two sticks at every step. Two different sequences have different behaviors. What is there to reconcile?
**
You’re missing the point. We can answer which; it just depends on which balls the gremlin takes out. That in turn determines how many.
I am not willing to say that there are an infinite number of balls in urn A. I’m not willing to say anything at all about the number of balls in urn A.
Saying anything about the number of balls in urn A after an infinite number of steps implicitly assumes that “the set of balls in urn A after an infinite number of steps” is a well-defined set. But it isn’t a well-defined set. If it were a well-defined set we’d be able to point at a particular ball and say “yes, that ball is in the set” or “no, that ball is not in the set”. Since we can’t make either of those statements for any of the balls, we have no basis for saying anything about the number of balls in urn A after an infinite number of steps.
And yes, that goes for urn B as well. Yes, after any finite number of steps there are two balls in the urn. But we can determine exactly which two balls those are. Since we have no way to determine which balls are or aren’t in urn B after an infinite number of steps, we cannot define “the set of balls in urn B after an infinite number of steps”, and therefore we have no basis to make any claim about the number of balls in urn B after an infinite number of steps.
Which goes right back to the point just made by Topologist: cardinality isn’t “continuous” in this fashion. There can be an infinite number of balls in the urn after every finite number of steps, but no balls in the urn after an infinite number of steps. Similarly, the fact that there are two balls in urn B after every finite number of steps doesn’t imply that we can say there are two balls in urn B after an infinite number of steps.
I agree. I don’t claim the answer to “how many” is zero, in your new setup. (I claim the answer to “how many” in your new setup is “the answer is not well-defined”.) However, your new setup is not the same as the setup in the OP, as I’ve already argued.
In the OP, we can answer the question of “which”. The answer to “which balls are remaining in the urn” in the OP is precisely “none”, because in the OP for every ball I can tell you the exact minute that the ball leaves the urn and never comes back. And since we know that each specific ball is not in the urn after an infinite number of steps, the urn must be empty at that “time”.
Regarding the sticks: I think there’s some confusion here. I don’t expect the stick lengths to converge as a sequence of sets at all. I expect the stick lengths to converge as a pair of sequences of numbers, which they do, and I expect the “set of points in each stick”, as a subset of the set of points in the whole, original stick to converge as a pair of sequences of sets, which they do.
I wrote “Just because you can’t answer “which” doesn’t mean the answer to “how many” is zero.” and I was trying to think if I could come up with the converse situation: one where I could name every ball in the urn, but where you (all of you, I guess) would still say the number of balls in the urn is zero. This is what I came up with:
At the start, the Gremlin has an infinite number of balls, with an infinite number of 1s, and infinite number of 2s and so on, and an empty urn. At the first step, he puts in a 1 and a 2, and then removes the 2 and discards it, never to be used again. At the second step, the gremlin adds a 1, replacinging the 1 that was in the urn, and also adds a 2. For convenience, I’ll call this the end of round 1.
I’ll specifiy that the time between steps is cut in half each time, so the supertask completes in finite time. Also, all balls removed from the urn are discarded, and never reused.
At the third step, the gremlin adds a 1, replacing the 1 in the urn, and also adds a 3. At the fourth step, the gremlin adds a 2, replacing the 2 in the urn, and also adds a 4. This ends round two.
Round three has steps 5 to 8, where the existing balls 1 to 4 are replaced, and balls 5 to 8 are added. This pattern is followed for subsequent rounds, with each round M having 2[sup]M-1[/sup] steps.
After each step N, balls 1 to N are present in the urn. My reading of the limit of the sequence of sets is that every number is represented in the limit set.
I guess I should be clear that I don’t think the urn is empty, just that I think that you will say the urn is empty, even though I can name an infinite number of balls in the urn.
ultrafilter wrote:
You can’t answer which for the case where the lower numbered ball is removed. Saying that “which” is “none” would be a circular argument.
How about why when the set of ball labels goes to the null set, this means there are no balls, but when the set of stick lengths goes to the null set, this doesn’t mean that there are no sticks.
orbifold wrote:
Does everyone else agree with this?
Urn B was specified to hold only two balls, no more. Do you agree that there are an infinite number of balls in the box? Where else can they be, except for in urn A?
There is always a lowest numbered ball left the urn. After the second step, there is always a second lowest ball. Which labeled ball that is may vary, but the nth-lowest is eventually always there.
Then you don’t understand the definitions we’re working with here. If every natural number is eventually present in the urn, then the limit set is N. But that’s a different scenario from the one in the OP, and it has a different outcome.
**
No it isn’t. If you believe so, please explain in great detail how it is. Any of the proofs I’ve presented should give you an example of the sort of detail that I’d like.
**
Because they’re different situations. If you change the assumptions, you can’t guarantee that the outcomes are different. The argument that because the set of stick lengths goes to the null set means there should be no sticks is just silly.
Furthermore, length is a very different beast from cardinality, and we’d probably be happier not considering it.
But the balls in this new scenario aren’t uniquely labelled. I’m sure you’re going to say “a ball numbered ‘1’ is in the urn”. But I’m just going to turn around and way “which ball numbered ‘1’?”
In this new scenario, saying that “a ball numbered ‘1’ is in the urn” does not imply that any given ball is in the urn. Look, suppose before you begin your new experiment I sneak in and add subscripts to all the labels, so that all the balls are now labelled uniquely. That is, there is now a ball 1[sub]1[/sub], 1[sub]2[/sub], and so on. After round 1, ball 1[sub]1[/sub] is in the urn. But in round two, that ball is removed and ball 1[sub]2[/sub] is put in its place. Ball 1[sub]1[/sub] is never seen again.
So once again I can point at any ball and tell you the exact minute it leaves the urn, never to return. And once again, you cannot name any single ball that does lie in the urn after an infinite number of steps. The best you can say is that “a ball numbered 1 is in the urn after every finite number of steps”, but that’s not good enough, because that’s not a statement about a specific ball.
So in the end, this new scenario is equivalent to the OP. The number of balls in the urn increases with each step, but for each ball there is a specific point at which the ball is removed, never to return. So after an infinite number of steps there will be no balls in the urn, because every one has been removed and disposed of.
For the same reason, this:
is also not a valid objection. The statement “after step n, there is always an nth-lowest ball in the urn” is not a statement about any individual ball, because the nth-lowest ball changes with every step. (Just as the “ball numbered 1” changes with every step in the above scenario.) Therefore this statement is not enough to imply that any particular ball is in the urn after an infinite number of steps.
How is it circular? Ball n gets removed at step n and is never put back. Therefore, ball n is not in the urn after an infinite number of steps. Since this is true for all n, every ball is not in the urn after an infinite number of steps. Therefore, the urn is empty after an infinite number of steps. How is that circular?
I explained this in an earlier post. The balls are individual, discrete objects drawn from a collection of all such objects and the labels uniquely identify each of these objects and distinguish them from one another. The sticks are not individual, discrete objects (since we’re explicitly taking them apart and mixing and matching the pieces) drawn from a collection of all sticks, and the lengths of the sticks are distinguishing the sticks in some larger collection from one another. Also, we’re not throwing sticks away at each step, never to be seen again; we’re just moving mass from one stick to another.
The proper balls-and-urns analogy to the sticks would be if the gremlin had an infinte number of balls and two urns, with all balls initially in the left urn, and just moved one ball from the left urn to the right urn at each step. Then after an infinite number of steps there would be no balls in the left urn and all the balls in the right urn. But there’s no reason to assume there are no urns.
Yes, there are an infinite number of balls in the box. For every ball, I can see that it is in the box at every step. Therefore, the set of balls in the box at each step forms a constant sequence of sets. So “the set of balls in the box after an infinite number of steps” is well-defined and equals the set of all balls. Therefore there are infinitely many balls in the box after infinitely many steps.
But the urn is another matter. For every ball, I can see that it is in urn A infinitely often and not in urn A infinitely often. Therefore the set of balls in urn A at each step forms a sequence of sets which does not converge. So “the set of balls in urn A after an infinite number of steps” is not well-defined. There is not a single ball for which I can state definitively that it is in urn A after infinitely many steps or not in urn A after infinitely many steps. Therefore, there is no conclusion to be drawn at all about the number of balls in urn A after infinitely many steps. The answer to “how many balls are in urn A after infinitely many steps” is undefined, even though the question is answerable for the box as a whole.
I’m now pretty sure that the answer is zero balls, almost every time. (What, you say I have too much time on my hands? No, not nearly enough, but I’m a sucker for certain kinds of puzzles.) That is to say, the empty set is the result with probability 1.
The argument is this: Let A[sub]n[/sub] be the collection of subsets of the natural numbers that contain n. As I mentioned earlier, the probability that n is not removed by the goblin is 0. This means that A[sub]n[/sub] has goblin measure 0 (the probability that the set of remaining balls is in
A[sub]n[/sub] is 0). A countable union of sets with measure 0 is another set with measure 0, so the union of all of the A[sub]n[/sub] has measure 0. But that union contains all nonempty subsets of the natural numbers. Hence the complement of that collection, the set containing only the empty set, has measure 1.
I thought sure you had made a statement that convergence to the null set meant the number of elements must be zero, but looking back, I see you didn’t. But if convergence to the null set doesn’t always mean “how many” equals zero, simply showing convergence to the null set doesn’t answer the question of “how many”. Although, I guess, we don’t even agree on whether this is actually the question being asked.
Orbifold wrote:
But now you’re looking for convergence of a different sequence.
I find this fascinating. We know there are an infinite number of balls in the box, and we know that of those, at most two of them are not in urn A, yet you say we don’t know that there are an infinite number of balls in urn A. I think this really highlights the disagreement.
What a mindfuck this thread is…
but I feel the need to chime in with my ignorant 2 cents…
ok. obviously you can not really define infinite time…
but to play along…
If an infinate time can be defined as (n)
the number of balls at time (n) will be equal to (2n)
If it assumes that at the exact time that you put the ball in, the gremlin would take one out not at the exact moment that you put the next 2 in, then it would be (2n) +1
but that is a pretty dumb answer… so I will ignore that…
that seemed to be all the OP was looking for.
If you want to know which balls are in the bucket , then you would look for
any integer in the range of (n)+1 and (2n)
ok that was my dumb answer…
and my REALLY dumb answer would be…
it depends how big the bucket is…