Under my model–convergence of sequences of sets–I can rigorously prove this.
Well, sure, but the contentious issue is “Why use convergence of sets rather than any other model?”.
Saying “I’m using convergence of sets” is the same thing as saying “I’m assuming that whenever ‘X is in cave Y’ is true after any sufficiently large number of finite steps, it is also true after infinitely many steps”. As Chronos says, you are making this assumption, while others might equally well want to make different assumptions, or simply say “The problem is undetermined”.
My post 31 offers what I believe is the most natural and most common way to approach this question. It is not the only way, so don’t complain if you disagree with me or otherwise want to quibble about anything other than a mathematical mistake I make.
After an hour, both the monkey and the ape have a countably infinite set of coconuts. So, in one sense, they both have the “same number” of coconuts, in the sense that there exists a bijection between their sets of coconuts.
After an hour, the monkey has a set of coconuts of density 1 in the set of natural numbers, and the ape has a set of coconuts of density 0 in the set of natural numbers. So, in another sense, the monkey has “all the coconuts” and the ape has “no coconuts”, in the sense of the natural density that I referenced previously. The ape still has infinitely many coconuts, though. Just very, very, very few (I am being non-rigorous here) compared to the monkey. If this is confusing, welcome to the study of measure theory and analytic number theory.
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Under my model–convergence of sequences of sets–I can rigorously prove this.
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Rigorously prove what? You need to define what you mean by convergence of sequences of sets. If you mean convergence in measure, with respect to the counting measure of the natural numbers, I think I’m on board with you here. But I’m not at all sure what you mean. If you mean something like
lim n -> infty S_n = S if and only if |S - S_n| -> 0 as n -> infty
then there are sets S where this doesn’t even make sense. I apologize if I am misunderstanding you here.
I assume “S_n converges to S” means “The characteristic functions of S_n converge pointwise to the characteristic function of S”, where the characteristic function of a set sends its elements to 1 and everything else to 0.
Hari are you saying that the description of the process determines which model of convergence is appropriate?
Such that the sequence 9991, 9992, 999*3, … has 0 as its limit because of the model of convergence that’s appropriate to this particular scenario, but that given another scenario, the very same sequence would have a different limit, because a different model of convergence would be appropriate in that case?
Ah, thank you for the clarification. If we add the assumption (which I assume you were making implicitly) that S_n is a subset of S for all n, then this is a very reasonable definition.
I see no reason why this approach is preferable to convergence in measure, where any measure satisfies the property of continuity from below, which gives Dr. Love the property he was looking for. But I freely admit that there are many ways to approach the problem, and none is inherently better than another. There is some lack of rigor in the thread, but it hasn’t stopped me from appreciating it at all.
I feel like adding that I am presently trying to show that some set of primes with some property has positive density in the set of all primes. Doing so would make a very hard problem using sieve techniques much easier. I may never succeed, but I console myself by reading threads like these, and knowing that no one else has solved the problem, either.
This. More intuitively, I’d say that S_n converges to S means that for any a \in S, there’s some N such that a \in S_n for all n > N, and likewise for b
otin S.
Let them do so. Once somebody lays out another model, we can discuss their relative merits. But right now, the objection seems to be that this can’t be the right model because it’s not the only possible one, and I just don’t find that compelling.
So by this definition, examining the scenario in my post 37, mangoes 1, 2, 3, etc. are all in the pipe. Likewise, if we ask what’s in any section N of the pipe, e.g. section 1, there is no mango in it by that definition. The same for sections 2, 3, etc.; all sections are empty.
That answer is inconsistent. Line up the mangoes 1, 2, 3, … just in front of the pipe. Then each step just consists of sliding the line of mangoes by one, further into the pipe. But this scenario can equally be described as moving the pipe onto the line of mangoes. The same scenario, just a different description.
For this description of the exact same scenario, section 1 is on the line of mangoes for all steps, and by your definition is on the line of mangoes at the end. Section 2 is likewise on the line of mangoes at the end. When we ask what section is on mango 1, it changes every step, and so by your definition there is no pipe on mango 1.
By symmetry, if mango 1 is in the pipe, section 1 has a mango. If section 1 is empty, mango 1 is not in the pipe. Your definition gives contradictory results, which depend on how the scenario is described, and so it is not correct.
There is an alternative model. See my post 35. I examine every action taken, and show that they all leave a coconut with a red 2 in the Ape’s cave. They all leave a coconut with red 3 in the Ape’s cave. There is no action that results in a coconut with a red 2 not present in the Ape’s cave. Once you’ve accounted for all the actions taken, that’s it.
Note that under this model, there is no contradiction. All mangoes are in the pipe. All sections have mangoes.
They do? Hari Seldon claims the ape has none. What is the bijection that maps the coconuts the ape has to the ones that the monkey has?
But isn’t your model also just convergence of functions from labels to caves, with a slightly different account of what caves have which labels at any moment? What makes your labelling system more “correct” than Hari’s original labelling system? What forces the universe to give red labels more respect than the original labels?
I think you aptly point out an ambiguity in the interpretation of the problem, but that your alternative model isn’t the One True Model either; there’s just this genuine ambiguity. The situation at the end is simply not specified until it’s, well, specified. Knowing the value of a function at some inputs doesn’t determine its value at any other inputs, except in the face of an appropriate continuity principle; and in this case, there hasn’t been a particular continuity principle specified. You could specify one, of course, but there isn’t a uniquely distinguished one clearly better than the others.
I don’t see how this problem can be considered the same as the original. In this case (as in Chronos’ Problem C) the monkey is “reserving” a set number of coconuts every time he enters the cave. I don’t have a problem with alternate formulations of the problem having indeterminate solutions.
I’ll state my justification more concretely : If, for a coconut x there exists a time T such that I can state where x is for all times t < T, and x is not moved for t >= T, then I know where it is. I’d like to go with a stronger version - if I know where x is at T, and it is not touched after that, then I know where it is for t > T. I’m prepared for either to be shown faulty. (Again, I don’t care that this leaves some of the problems given in this thread unsolvable.)
As an aside in both these problems, it’s not clear which ‘original label’ the monkey takes. In Chronos C, it seems that the coconut the ape has, if it exists, has the label 1. In yours, it’s not clear if the monkey is taking the coconut originally labeled 2 or 1001, though I take you to mean 1001.
And to clarify my variation given above - in my approach there’s a difference between the cases where the monkey continues to steal the lowest-numbered coconut in the cave, or the natural number for that time.
I’m just adding additional red numbers to the coconuts. (I guess I’m implicitly assuming the original numbers are black.) Everything occurs just as it does in the original. If the monkey takes the coconut labeled 2 in the original, he takes the coconut labeled with black 2. He never changes the black numbers.
The red numbers are just for keeping track of the coconuts the Ape has in an explicit manner.
Hari Seldon is wrong. In any of the intervals of time in the problem (30 min., 15 min., 7.5 min., 3.75 min., etc.) the ape is adding 999 coconuts to his pile. There is no way that that his pile of coconuts approaches zero as time approaches 1 hour. His pile of coconuts grows without bound as a function of time.
The monkey’s pile of coconuts also grows without bound as a function of time. In the limit, they both have a countably infinite pile of coconuts. As has been pointed out previously, there exists a bijection between such sets.
The difference, which I have pointed out, is that the asymptotic density of the monkey’s coconuts is 1, while the asymptotic density of the ape’s coconuts is zero.
Hari Seldon is of course correct that given any coconut, labeled, say, n, there exists some time interval, say t_n, such that the monkey takes coconut n. That is why the most reasonable interpretation (I feel) is the density consideration: compare the (size of the) sets of coconuts in the nth unit of time to the size of the natural numbers less than n as the number of units of time approaches infinity. This is getting at the heart of the whole matter here! How do we compare countably infinite sets? This comes up all the time in mathematics. I find it hard to believe that none of the other (pure) mathematicians have had occasion to make such considerations before. Sets of measure zero are wonderful pathological cases in measure theory, topology, analysis, etc.
RayMan, at some point the monkey takes the coconut with (Hari-assigned) label 1. When does the ape get it back? Or, if the ape never gets it back, what coconuts does the ape end up with and when does it get those?
Furthermore, you speak of asymptotic density (i.e., the limiting number of coconuts with label < N in each particular cave), but this density is extremely sensitive to the labelling you use. Do you think the answer depends on how you choose to label the coconuts? What if they simultaneously have different labels (a la ZenBeam’s black label vs. red label distinction) which would lead to different results?
There’s no clear right or wrong answer here. It’s just a matter of what, if any, bridge principle you choose for how the value of the “What is the world like after t hours?” function’s value at t = 1 is related to its values at [0, 1). And, as we’ve all seen, different (but in each case, very, very natural) bridge principles give contradictory results. You can’t have them all at once. That’s the paradox.
Suppose we played the following rather similar game:
Every minute, a new baby is born and lines up at the back of the queue at Hell’s DMV. Every hour, on the hour, the person at the front of the line finally receives their driver’s license and gets to drive off to Heaven.
Each baby, the moment it’s born, knows exactly how long it’ll take to get its driver license; it knows it will leave someday. There’s only so many people ahead of it in line, after all. There’s never any doubt in that baby’s mind that it will make it out; it even knows the exact date.
And that’s right, isn’t it? Every baby does leave at some point, right on schedule.
But now fast forward, past one day, past two days, past three days, past three years, past three centuries, past three millenia, onward and onward, to after every finite number of days has passed… What’s the world look like then?
“You twit”, you say. “There’s no such thing as after every finite number of days”.
Who says? But ok, we’ll play it your way. Just relabel the calendar; speed things up purely nominally. What was k days after the start on the Old Style calendar will be 1 - 1/2^k days after the start on the New Style calendar.
Now what’s the situation like after 1 day (N.S.)? Are you really able to figure it out just because we re-named the calendar?
What if instead of different days in time, we were talking about different locations in space? Would you then feign to know what things are like a mile to the right, just from knowing what things are like less than a mile to the right?
Let me put it this way: What we’re quibbling over isn’t math. We’re quibbling over, say, physics: how does the world behave? What principle correctly describes how the world’s state at time t is related to its state over the times prior to t? Which is either an empirical question (if we’re talking about the real world) or just a question of what particular sort of story we’re interested in writing (if we don’t care about the real world, per se). But is not, in any case, a question for which there is some mathematical means to distinguish between correct and incorrect answers. The issue is extramathematical. The mathematics is all perfectly straightforward (clearly, the black labels converge to such and such, and clearly, the red labels converge to such and such, and the measures act like such and such, and so on, and there’s no contradiction in any of that).
Can we write functions that describe the size of the ape pile, A(n), and the size of the monkey pile, M(n), at the end of the nth interval?
A(n)=999n or A(n)=1000n-M(n)
M(n)=n
Anything wrong about this? We’re claiming that A(n)->0 as n->infinity?
Clearly, as n grows large, A(n) also grows unboundedly large. That is not in any dispute. What’s in dispute is the actual value of A at n = infinity, so to speak. Does it continuously match this infinite limit? Or is it discontinuously zero in order to make the coconuts’ position functions continuous? It’s not possible for everything to be continuous that we’d want to be continuous; if A is continuous, something else must be discontinuous.
Could you restate what the coconuts’ positions functions are? Just which pile a certain coconut is in, if any? Like:
P2(n)=
wild if n=0
Ape if n=1
Monkey if n>1
So if n=infinity, the position function for each coconut would indicate that they’re in the Monkey’s pile… ok.
So, math just says “well, them’s the breaks” in this situation? It just depends on whatever we feel is more important to fit our intuition?
The ape never gets any coconuts back. We consider the number of coconuts the ape and monkey have as the number of units of time approach infinity.
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Furthermore, you speak of asymptotic density (i.e., the limiting number of coconuts with label < N in each particular cave), but this density is extremely sensitive to the labelling you use. Do you think the answer depends on how you choose to label the coconuts? What if they simultaneously have different labels (a la ZenBeam’s black label vs. red label distinction) which would lead to different results?
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You are correct that labeling is relevant to the problem as posed. If the monkey were to take from the ape a set of coconuts not of density one, well, he would end up with a set of coconuts…not of density one. For example, if the monkey stole the coconut labeled as the nth prime in the nth unit of time, he would end up with a set of coconuts of density zero, and the ape, having the set of coconuts labeled with the integers lying in the complement of the primes in the natural numbers, would have a set of coconuts of density one.
I can see your objection already: the set of primes and the set of natural numbers, both being countably infinite, lie in one-to-one correspondence! It’s really taking the same thing! This is where we return to the concept of natural density. The primes have density zero in the set of natural numbers. There are, in some well defined and widely accepted sense (the sense of natural density) “almost no” primes in the set of natural numbers. Depending on how we label the coconuts, the monkey may “leave behind” enough coconuts that the ape has a set of positive density.
Everyone else types faster than me. I see there have been several replies since I began my own. I feel I should make a few points:
It doesn’t make sense to talk about what happens at time t=1 hour. We can only describe the relationship between the sets as time approaches 1 hour asymptotically.
I have said repeatedly in this thread that there is more than one way to approach this problem, and no approach is inherently better than any other.