# Math paradox: putting infinite number of balls in a box **URL:** \ **Category:** Factual Questions\ **Created:** [September 5, 2018, 11:15pm UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841 "2018-09-05T23:15:55Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Joey\_P](https://avatars.discourse-cdn.com/v4/letter/j/919ad9/32.png) [@Joey\_P](https://boards.straightdope.com/u/Joey_P)\ **Post date:** [September 6, 2018, 3:08am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/21 "2018-09-06T03:08:31Z")
> [@RaftPeople](#): > > I think there are multiple issues: > 1 - that you can complete infinite tasks in a finite amount of time (if so, what was the number of the third to last ball placed in the box?) > > 2 - that you can describe a problem using a formula based on the state at discrete time steps, and then ask for the state at a nonsensical time step (there is no time step called infinity) 1)The paradox, as stated in the OP, had no time limit, it goes on forever. That’s sort of the nature of formulas that create asymptotes. 2)Again, I never mentioned a time or time limit, only how many steps have been done, N. There is a difference, especially depending on how the question is worded. 1 & 2)The time issue is addressed in both the Zeno and Ross-Littlewood paradox. Basically, whatever time you want this to finish at, it can only _approach_ that time since you can’t do (as you mentioned) an infinite amount of operations in a finite amount of time. For times less than the time it’s scheduled to be finished at, it would either change the math to be infinity or have an actual, numerical answer. However, since the OP didn’t specify how long each operation takes, we can’t do anything with that aspect of it and it’s irrelevant to the question as stated. It’s like saying “I’m going to drive from L.A. to Santa Fe, how long will it take?”, without knowing the velocity, it can’t be answered. ---
**Author:** ![Joey\_P](https://avatars.discourse-cdn.com/v4/letter/j/919ad9/32.png) [@Joey\_P](https://boards.straightdope.com/u/Joey_P)\ **Post date:** [September 6, 2018, 3:10am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/22 "2018-09-06T03:10:48Z")
> [@Thudlow\_Boink](#): > > It seems to me (without reading up on it to see if this line of reasoning has been pursued) that what we have here is an infinite sequence of _sets_ (i.e. A[sub]n[/sub] = the set of balls in the box after step n), and the question is, does this sequence converge? What is the limit of the sequence as n approaches infinity? > > So, what _does_ it mean for a sequence of sets to converge? Do we have to have a metric defined on the set of sets of balls? Barf…years ago I started a thread on that. From time to time I look back at it and it’s all greek to me at this point. Hell, I think it was all greek to me at that point as well. Here it is, I’m not even going to attempt to read it. > **[Help with Pointwise Convergence.](https://boards.straightdope.com/sdmb/showthread.php?t=197603)** > > Yes this is a homework problem. I’ve been racking my brain for a few days now trying to get a grasp on the concept of pointwise convergence. I really don’t care for my textbook much, as it really isn’t very good as far as decent examples are... ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [September 6, 2018, 3:50am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/23 "2018-09-06T03:50:45Z")
I disagree that there is no limit. This is an infinite sequence. Simply looking at step N when N is really large can give us an idea. No matter how large N is, there are always at least 9N balls in the system. There is never a point where there are 0 balls. And the sequence increases without bound. Hence the limit is infinity, and thus the answer to the OP’s question is that there are still an infinite number of balls, not 0. Now, granted, one can say that infinity means “no limit,” but, in math, an infinite limit is possible. We can say that the limit of f(N) as N approaches infinity is infinity. ---
**Author:** ![RaftPeople](https://avatars.discourse-cdn.com/v4/letter/r/6f9a4e/32.png) [@RaftPeople](https://boards.straightdope.com/u/RaftPeople)\ **Post date:** [September 6, 2018, 4:10am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/24 "2018-09-06T04:10:44Z")
> [@Joey\_P](#): > > It’s like saying “I’m going to drive from L.A. to Santa Fe, how long will it take?”, without knowing the velocity, it can’t be answered. But we can set an upper bound on the number of steps possible if we assume each step consumes the smallest amount of energy possible (plank?) which renders the question nonsense. I get that limits in math are useful, but this problem seems to want to switch between finite concrete questions and answers and limit based math. ---
**Author:** ![Joey\_P](https://avatars.discourse-cdn.com/v4/letter/j/919ad9/32.png) [@Joey\_P](https://boards.straightdope.com/u/Joey_P)\ **Post date:** [September 6, 2018, 4:31am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/25 "2018-09-06T04:31:54Z")
> [@RaftPeople](#): > > But we can set an upper bound on the number of steps possible if we assume each step consumes the smallest amount of energy possible (plank?) which renders the question nonsense. > > I get that limits in math are useful, but this problem seems to want to switch between finite concrete questions and answers and limit based math. The OP stated that step N should be “repeated for all natural numbers”. ISTM, if you set an upper bound on the number of steps, you’re changing the question. Furthermore, if you set an upper bound on N, there’s no question there are still balls left in the box and it’s easy to calculate the number. I’m not sure how energy consumption comes into play. As far as limits, that’s one of the first tools given to us in calculus to deal with a variable that continues towards, but never reaches a given number (or infinity). It’s a perfectly cromulant method that’s been in use for nearly 2500 years (or 400, depending on who invented it). Even if you ignore that and hand it to a high school Algebra student with no concept of limits, I don’t see how they’d come up with zero as the final answer. Also, regarding switching between finite concrete answers and limit based math, again, that’s part of calculus. Plug some numbers in, plug more numbers in and eventually find an equation to interpolate/extrapolate what happens elsewhere. ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [September 6, 2018, 4:59am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/26 "2018-09-06T04:59:24Z")
> [@Joey\_P](#): > > There’s a difference between “every single ball that you can name has been removed” and the actual case which would be “every single ball that you can name either has been or will be removed”. It the 'will be" that changes everything. You alluded to it when you said “there are vast hordes of larger numbered balls in the box”. What if I name one of the balls that are still int he vast hordes? Your statement only works if, instead of removing the lowest, you remove a random ball. I think you must be misunderstanding the OP. We’re supposing that we’re looking at the box after ALL steps have been completed. There are no steps that “will be” done in the future. My “vast hordes” statement referred to the situation at each step, while you were tediously putting balls in and out of the box. The question is what’s in the box when all these steps are finished. It’s possible to object to the idea that it’s meaningful to consider a situation after an infinite number of steps, as beowulff and Chronos and others have done in different ways. But if you accept that the whole countable infinite number of steps have been done as the OP states, then for any number N, ball N has been removed. Look, if there is a ball in the box, it has a number on it, right? Call it N. But that ball was removed in step N. Therefore there can be no ball in the box. QED. ---
**Author:** ![harmonicamoon](https://avatars.discourse-cdn.com/v4/letter/h/65b543/32.png) [@harmonicamoon](https://boards.straightdope.com/u/harmonicamoon)\ **Post date:** [September 6, 2018, 5:22am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/27 "2018-09-06T05:22:24Z")
You folks are missing the larger problem. What size box holds an infinite amount of balls? If it is an infinite box, is it still called a box? ---
**Author:** ![Riemann](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/riemann/32/3133_2.png) [@Riemann](https://boards.straightdope.com/u/Riemann)\ **Post date:** [September 6, 2018, 5:28am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/28 "2018-09-06T05:28:44Z")
> [@harmonicamoon](#): > > What size box holds an infinite amount of balls? Yo mama’s, obviously. ---
**Author:** ![Malden\_Capell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/malden_capell/32/5274_2.png) [@Malden\_Capell](https://boards.straightdope.com/u/Malden_Capell)\ **Post date:** [September 6, 2018, 5:44am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/29 "2018-09-06T05:44:43Z")
I have to be that guy, but what is the point of exercises like this? Infinity isn’t possible for things like balls, boxes, or numbers of steps a person can operate, and as it never stops, no answer is possible. So is it answerable at all, or even worth answering? Sent from my iPad using Tapatalk ---
**Author:** ![Joey\_P](https://avatars.discourse-cdn.com/v4/letter/j/919ad9/32.png) [@Joey\_P](https://boards.straightdope.com/u/Joey_P)\ **Post date:** [September 6, 2018, 6:01am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/30 "2018-09-06T06:01:34Z")
> [@harmonicamoon](#): > > You folks are missing the larger problem. > > What size box holds an infinite amount of balls? > > If it is an infinite box, is it still called a box? It can hold an infinite about of balls if they are infinitesimally small and/or the box is infinitely big. > [@Malden\_Capell](#): > > I have to be that guy, but what is the point of exercises like this? Infinity isn’t possible for things like balls, boxes, or numbers of steps a person can operate, and as it never stops, no answer is possible. > > So is it answerable at all, or even worth answering? These types of math problems (barring the paradox thing) are the fundamentals of calculus. One of the first things you learn in calculus is finding the area under a curve. It’s done by creating a [series of rectangular columns](https://www.intmath.com/blog/wp-content/images/2014/06/area-under-curve-rectangles.png) and adding their areas and [making them smaller](http://mathandmultimedia.com/wp-content/uploads/2012/10/area-under-a-curve.jpg) (and therefore a more accurate representation of the curve) until they are infinitely small. In fact, the wiki’s intro on their calc page says “Calculus has historically been called “the calculus of infinitesimals”” That is, at least in part, what makes calculus calculus. For this reason, exercises about an infinite number of something in something are very common. Deconstructed like that, they seem obscure and pointless, but as you learn to work with them they really do have a lot of real world applications. For example. if you take a parabola that goes from -1 to 1 with a magnitude of 1 and find the area under that, then rotate the curve about the X axis (like finding the area of a rectangle, then extending it upwards), you have the volume of a sphere. I like to say that in algebra you were given the formula for a sphere, in calculus you derive it. ---
**Author:** ![Malden\_Capell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/malden_capell/32/5274_2.png) [@Malden\_Capell](https://boards.straightdope.com/u/Malden_Capell)\ **Post date:** [September 6, 2018, 6:27am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/31 "2018-09-06T06:27:52Z")
And what can you do with this sphere that’s been made accidentally? Sent from my iPad using Tapatalk ---
**Author:** ![Saffer](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saffer/32/6440_2.png) [@Saffer](https://boards.straightdope.com/u/Saffer)\ **Post date:** [September 6, 2018, 7:25am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/32 "2018-09-06T07:25:44Z")
> [@Malden\_Capell](#): > > And what can you do with this sphere that’s been made accidentally? > Sent from my iPad using Tapatalk You could calculate the volume of balls in a box. Sent from my iPhone using Tapatalk ---
**Author:** ![hogarth](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hogarth/32/1773_2.png) [@hogarth](https://boards.straightdope.com/u/hogarth)\ **Post date:** [September 6, 2018, 11:39am UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/33 "2018-09-06T11:39:34Z")
The series diverges. To say that a divergent series is equal to “infinity” or “zero” or whatever is misleading in this context. > **[Divergent series](https://en.wikipedia.org/wiki/Divergent_series)** > > Les séries divergentes sont en général quelque chose de bien fatal et c’est une honte qu’on ose y fonder aucune démonstration. > ("Divergent series are in general something fatal, and it is a disgrace to base any proof on them." Often translated as "Divergent series are an invention of the devil …") > N. H. Abel, letter to Holmboe, January 1826, reprinted in volume 2 of his collected papers. > In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite s... ---
**Author:** ![Quercus](https://avatars.discourse-cdn.com/v4/letter/q/7ab992/32.png) [@Quercus](https://boards.straightdope.com/u/Quercus)\ **Post date:** [September 6, 2018, 12:57pm UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/34 "2018-09-06T12:57:27Z")
> [@Itself](#): > > Put \*X[SUB]n/SUB \*= 1 if ball _n_ is in the box at time _t_ and 0 otherwise. The total number of balls in the box at a fixed time _t_ is then just \sum X[SUB]_n_/SUB, the sum of the _X[SUB]n/SUB_. The paradox then becomes the statement that lim[SUB]t-\>∞[/SUB] \sum[SUB]_n_[/SUB] _X[SUB]n/SUB_ and \sum lim[SUB]t-\>∞[/SUB] _X[SUB]n/SUB_ differ, but that’s hardly surprising; infinite sums don’t commute with limits in general. > > (Also, trying to typeset math here sucks.) Yes. The ‘paradox’ is not a math issue; it’s about how using English words to describe math problems (especially ones with infinity) can lead to confusion. [In other words, like the majority of arguments on the internet, this is an argument about what a word means, not about any real-world or mathematical facts]. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [September 6, 2018, 1:10pm UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/35 "2018-09-06T13:10:02Z")
> [@markn\_1](#): > > 1. If you say there are an infinite number of balls, name one. For any number N, if you claim that ball N is in the box, I say no, it was removed at step N. There are an infinite number of balls, but the specific balls keep changing. Consider two other scenarios: Scenario A At step 1, put ball 1 in the box. At step 2, replace it with balls 2 and 3. At step 3, replace them with balls 4, 5, and 6. … At step n, replace whatever’s currently in the box with the next n balls in line. Scenario B On any given day, some people are born, and some people die. Assume for the sake of argument that, on every particular day, more people are born than die, and that every individual person will eventually die. In all three scenarios, the number of balls/people keeps growing without bound; but no one gets to stick around forever. ---
**Author:** ![Patch](https://avatars.discourse-cdn.com/v4/letter/p/a183cd/32.png) [@Patch](https://boards.straightdope.com/u/Patch)\ **Post date:** [September 6, 2018, 1:22pm UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/36 "2018-09-06T13:22:40Z")
> [@Joey\_P](#): > > …the box is infinitely big. Don’t tell Amazon about that box! They’ll use it to ship a microSD card. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [September 6, 2018, 1:28pm UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/37 "2018-09-06T13:28:27Z")
> [@hogarth](#): > > The series diverges. To say that a divergent series is equal to “infinity” or “zero” or whatever is misleading in this context. > > [Divergent series - Wikipedia](https://en.wikipedia.org/wiki/Divergent_series) What series? There really is not much to add to the discussion of this paradox that has not been covered above or in the Wikipedia article, but it is worth re-iterating that the tricky part is to define a limiting step, call it step ω, that describes the contents of the box after a countably infinite number of steps; this is not explicitly specified in the OP or in Littlewood’s book, though Littlewood takes it for granted the reader knows what he is talking about. If we insist, how are we to make it explicit? Call the set of balls in the box after n steps A[sub]n[/sub]. Then A₁ = { 2, 3, 4, 5, 6, 7, 8, 9, 10 }, A₂ = { 3, 4, …, 20 }, etc. If we want to define some sort of limit it is certainly reasonable to consider ⋃[sub]n≥1[/sub]⋂[sub]k≥n[/sub]A[sub]k[/sub] and ⋂[sub]n≥1[/sub]⋃[sub]k≥n[/sub]A[sub]k[/sub], assuming they coincide, that is, in the case when every ball either eventually remains in the box and is not removed or remains outside the box and is never again put in. This is the case for the A[sub]n[/sub] defined by Littlewood, because ball number n is absent after step n, therefore A[sub]ω[/sub] = lim[sub]n→∞[/sub]A[sub]n[/sub] exists and is empty. ETA As Littlewood says, “An analyst is constantly meeting just such things… and without noticing anything paradoxical.” ---
**Author:** ![Joey\_P](https://avatars.discourse-cdn.com/v4/letter/j/919ad9/32.png) [@Joey\_P](https://boards.straightdope.com/u/Joey_P)\ **Post date:** [September 6, 2018, 1:43pm UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/38 "2018-09-06T13:43:20Z")
> [@Malden\_Capell](#): > > And what can you do with this sphere that’s been made accidentally? > Sent from my iPad using Tapatalk The sphere was just an example. You asked what the point was of these types of things, I showing that it’s the basis for calculus, not just a random exercise. ---
**Author:** ![Lucas\_Jackson](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lucas_jackson/32/303_2.png) [@Lucas\_Jackson](https://boards.straightdope.com/u/Lucas_Jackson)\ **Post date:** [September 6, 2018, 1:56pm UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/39 "2018-09-06T13:56:53Z")
> [@Patch](#): > > Don’t tell Amazon about that box! They’ll use it to ship a microSD card. [clicks ‘like’ box] ---
**Author:** ![Jasmine](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/jasmine/32/2964_2.png) [@Jasmine](https://boards.straightdope.com/u/Jasmine)\ **Post date:** [September 6, 2018, 2:00pm UTC](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841/40 "2018-09-06T14:00:18Z")
> [@beowulff](#): > > Well, there is no end, so that makes the question moot. Exactly. > [@](#): > > And so on. In the limit, after you have performed step N for all natural numbers N, what is in the box? The question itself is faulty. [Previous page](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841.md?page=1) [Next page](https://boards.straightdope.com/t/math-paradox-putting-infinite-number-of-balls-in-a-box/820841.md?page=3)