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.
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.
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.
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.
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.
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?
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It can hold an infinite about of balls if they are infinitesimally small and/or the box is infinitely big.
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 and adding their areas and making them smaller (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.
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].
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.
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.”
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.
