You are infinitely approaching the “Don’t be a jerk”, mode.
::shrug:: I thought it was funny, and not meant to be taken personally.
The difference between the area under a curve and the OP is that in the OP the infinite items are given explicit unique identifiers, and we are then asked to counter logic applied to those unique identifiers.
In calculus, you don’t uniquely identify each discrete chunk of area and then reason about those discrete elements after you’ve conceptually transformed the entire set into a different animal using limits.
This is correct.
The real mathematicians will correct me if I’m wrong, but I believe this is incorrect. The cardinality of the infinite set of rational numbers is equal to the cardinality of the infinite set of the natural numbers, i.e. there exists a one-to-one mapping between the two sets. Just don’t ask me to prove it - this is the point at which I abandoned my mathematical studies and switched to philosophy, which is really what this thread is about :).
The cardinality of the infinite set of the Real numbers, however, is indeed greater than the cardinality of the infinite set of the natural numbers, as proved by Cantor’s diagonal argument.
True, but that’s why I mentioned (earlier) that IMO, requiring the balls to be numbered is what changes this from a paradox to a riddle. ISTM, it would make more sense if numbering them was part of the argument that there are zero balls in the box as opposed.
The area under a curve, as I’ve stated already, has nothing to do with this other than trying to explain to Malden Capell that pushing things towards infinity isn’t a useless exercise but rather a fundamental part of calculus.
Easy for you to say. He ain’t talking’ about jo mama.
I thought it was quite crude.
You are correct and well done with a reasonable definition for what constitutes the ‘same’ infinity.
It is true we can have ‘different’ infinities that have distinct and interesting properties. The particular example given by etasyde is not really such a one.
One way we can categorize infinities is, as you note, by their cardinality. Whole numbers and the “half-numbers” as defined by etasyde have the same cardinality. They can be placed in a 1-1 correspondence with each other, i.e. they are the “same” kind of infinity in some (well defined by not intuitive to humans) sense.
Another way to look at the removal of each named ball, is when one gets to the ‘end’ (infinity), you have moved all the named balls, yes every one, and you can’t count higher. But since you have replaced them with 10x more balls of greater values, now you only will have balls that really don’t have defined names anymore, infinity +1, Infinity +2, etc, which can not be a named ball as it is undefined, yet their they are cluttering up the box.
I don’t know if anyone will find value in this post but I’m posting it anyway.
I’m going to restate the problem in a more abstract way to get rid of all the distractions about balls and boxes and time limits (or monkeys and coconuts for those familiar with the problem in a different context). This will (hopefully) make it clear that what we’re really dealing with here is the difference between pointwise and uniform convergence.
Define a sequence of functions f[sub]n/sub for n = 1, 2, 3, … piecewise as follows: f[sub]n/sub = 1 for n <= x < 10 and f[sub]n/sub = 0 otherwise.
Note that the integral of f[sub]n/sub over the real line is equal to 9n.
Now the limit of f[sub]n/sub as n goes to infinity is f(x) = 0. However, this convergence is pointwise but not uniform. (Left as an exercise for the reader.)
As a result of this the limit of the integral of f[sub]n/sub as n goes to infinity grows without bound while the integral of the limit of f[sub]n/sub is zero.
Hope this helps.
Ah the paradoxes of the infinite. Certainly at the “end” no ball is in the box, since very ball is removed. No question of that. And while there are different orders of infinity, they do not enter this question. For a 1-1 correspondence between the positive rationals and positive integers, write down all the rationals in lowest terms in the following order: first write down all those for which the sum of numerator and denominator is 2. There is only one, namely 1/1. Then all those with sum 3, in order of the numerator: 1/2, 2/1. Then 4: 1/3, 3/1 (omit 2/2). Then 5: 1/4, 2/3, 3/2, 4/1. Continue to get a series 1/1,1/2,2/1,1/3,3/1,1/4,2/3,3/2,4/1,1/5,5/1,1/6,2/5,3/4,4/3,5/2,6/1,… The nth term of this sequence corresponds to n. I once saw an actual formula for a different pairing, but I have forgotten it.
IMHO, a useful conceptualization of a countable infinity is that of a dynamic set that is always increasing. Thus the stated problem trivially becomes a non-paradox because the requisite condition “at the end” does not exist. This differs from a meaningful solution in integral calculus because there, as the number of elements being summed tends to infinity, their magnitude tends to zero according to some function, so the solution converges to a limit.
That gets you most of the way to my favorite enumeration of the (positive) rationals.
Yup, very cool. Remarkable that such a simple procedure has this property:
Ah, the Stern-Brocot tree. Has all sorts of connections to Fibonacci numbers, e, and ton of other things. I highly recommend the section on it in Graham, Knuth,and Patashnik’s Concrete Mathematics. But not a lot to do with the OP.
Another way to attack the poor definition of the problem is to ask what you are doing when the number of balls approaches 10[sup]80[/sup]. How do you label the balls? How do you find the ball to remove? How long is it taking you to remove the ball? What the hey is this box made of? How big is it? Esp. when you really don’t want it to collapse into a black hole. Etc. 
I feel like throwing in a theoretical note for the calculus/analysis nerds, that may or may not have any real relevance to the subject of this thread.
When we talk Calculus, we often use verbs like “approaches,” “tends to,” converges," or “making them smaller,” and talk in terms of process or movement. This is done for convenience and, perhaps, to help our intuition. But it can possibly lead us astray if it makes us think about how long the process takes or what happens in the middle or at the end of it. Because the standard modern, Weierstrassian definition of a limit doesn’t use such concepts.
Here’s how William Dunham described the difference between Cauchy’s earlier definition of a limit and Weierstrass’s:
If you can ask “What is the number of a ball in the box?”, then I can ask “At what step did the number of balls in the box decrease?”.
Would it help if we added a treadmill?
No, but please, please, please, don’t put the box with the balls behind a door and put some goats behind other doors!!!
That only works with an infinite number of doors and an infinite number of goats.
I don’t think “eventually” works with an infinite number of steps.
Let’s say steps 2-Infinity are changed so that you add 1 ball instead of 10 while taking away 1. The box has 9 balls, and it will stay fixed at 9 forever, despite the same claim from above that you will remove every single ball.
You can, instead of having one large box, have each step add a box with 10 balls, then move the lowest numbered ball to the previous box, cascading down to the first box. Each box will therefore be fixed at 9 balls, and the number of boxes will increase without bound.