So, imagine you have an infinitely large bucket and an infinite number of balls labelled 1, 2, 3, etc. At 1 minute to 12 am, you put balls 1-10 into the bucket and remove ball 10. At 1/2 minute to 12, you put in balls 11-20 and remove ball 20. At 1/4 of a minute to 12, you put in balls 21-30 and remove ball 30. You continue doing this until 12. At 12, how many balls are in the bucket? It should be pretty clear that an infinite number of balls are in the bucket. Specifically those balls that are not multiples of 10.

Now, let's do it again, but change the problem a little bit. Instead of removing balls 10, 20, 30, etc. we will remove balls 1, 2, 3, etc. Everything else remains the same. Now how many balls are in the bucket at 12? This time the answer is none. Why? Because every ball that gets put in also gets removed. Ball n gets removed 2^(1-n) minutes before 12. Since, for every ball we can find a time before 12 at which it is removed, there cannot be any balls in the bucket at 12.

Seems rather paradoxical since at any one step of the process you're increasing the number of balls by 9, so how could you ever empty the bucket? Thus is the nature of infinity. And, just to make this a little more interesting, what if, instead of removing a specific ball you remove a random ball chosen uniformly from the balls that are already in the bucket? How many balls are in the bucket at 12 then?

Rollback Post to RevisionRollBack

Never attribute to malice what can adequately be explained by incompetence.

So, imagine you have an infinitely large bucket and an infinite number of balls labelled 1, 2, 3, etc. At 1 minute to 12 am, you put balls 1-10 into the bucket and remove ball 10. ... tldr

At 12 there are an infinite number of balls added, and an infinite number of balls removed, but since the number of balls added and removed are not equal, you cannot say it is zero. comparing one infinity to another by itself is meaningless. Instead, you have to compare the functions that produced those infinities.

Using l'Hôpital's rule, you can figure that as the time approaches 12, the number of balls approaches infinity, not zero.

I think this has been stated on many of the threads based around math problems here before it. Infinite is not a number, it's a concept. Therefore, infinite cannot be used in the equation.

If you think about it like like this:
n = step = infinite
balls = n*10-1 = infinite

(In actual fact, your problem, even the second one, would never spit out 0 because you never reach infinite therefore you continue to the next step. So other than the empty bucket at 0, the minimum number of balls ever in the bucket at one time is 9)

I never actually performed any arithmetic operations on infinities. All I did was see what balls would be left in the bucket at 12. In the first example, all balls that aren't multiples of 10 are left. There's an infinite number of those. In the second example, there can't possibly be any balls left since any given ball is removed at some point. If, as some of you claim, there are an infinite number of balls left (or any, for that matter), you should be able to name the numbers that are on these balls (as I did in the first example).

Of course the example is weird, that's the point, but there's nothing actually mathematically invalid about either case. You can't physically realize this scenario, but that's true of a lot of things in mathematics. Another good example is the Banach-Tarski paradox which shows that you can take a sphere, cut it into a finite number of pieces and then through only translations and rotations assemble those pieces into two spheres identical to the first. The reason that this one isn't physically realizable is because the pieces you chop the sphere into are not measurable (I'm not going to go into measure theory here, since I apparently get a tl;dr after a mere 5 lines, if I could relate it to boobs, I'm sure it'd go over better).

The point is that particular sets of axioms don't always lead to intuitively correct results (but that doesn't mean they're incorrect if they follow consistently from the axioms), a lot of other important results in mathematics rely on these axioms (such as the axiom of choice), so usually mathematicians just accept some of the weirdness.

Rollback Post to RevisionRollBack

Never attribute to malice what can adequately be explained by incompetence.

Clearly infinite for both cases. Even in the second one, the number of balls added is 9 times the number removed. While you do add and remove infinitely many balls, this is one time when you can say you are adding more infinitely many balls than you are removing. One infinity is in a way "bigger" than the other.

You could also look at it as always adding balls and instead of removing them you add "negative balls". This gives the equation infinity - infinity, which does not equal zero (can, but not here).

One infinity bigger than another infinity? Do you realize what you're saying?

Explanation of a concept, it need not be mathematically accurate, it only needs to represent the concept of their sizes.

Infinity minus infinity is always undefined. They're not numbers. You can't do infinity - infinity the same way you can't do horse - potato. It doesn't make any sense. I'm also flabbergasted by your assumption that inf - inf can equal different things in different situations.

check it out:

say | inf - inf = 0
then | inf - inf + 1 = 1
given that | inf + 1 = inf
then | inf - inf = 1

It don't work. Inf - inf is undefined.

I notice how you say you can't do it, then prove it by doing it. For instance, both x and 2x^2 both approach infinity as x does. But at any given finite elapsation (that a word?) of x, the second will always be larger. That is what I am saying above. After any finite iteration in the problem above (the second one from OP) there will always be more balls in the bucket than have been removed. If we iterate towards infinity, the difference between them is infinity, but at the same time an infinite number of balls have been removed, and an infinite number have been added.

I am also flabbergasted that you only look at the symbols and not the concepts.

They're both countable infinities, which is a measure of their cardinality. They are not the same size, since for every element in one, there's exactly one element in the other. All the infinities are countable in this problem and, therefore, they all have the same "size".

Clearly infinite for both cases.

If that truly is the case, then you should be able to tell me which balls are in the bucket. In the first case that's easy, the balls that are in the bucket are the balls that aren't a multiple of 10. Which balls are still left in the bucket in the second case?

If we iterate towards infinity, the difference between them is infinity

That's true if you ignore a fundamental structure that we imposed on the problem: the fact that the balls are unique. Try imagining the problem as being broken into two steps. First we fill the bucket with all the balls 10 at a time and we don't remove any. Then, as an entirely separate series of steps, we remove the balls 1 at a time starting with the least numbered ball. The only thing different now is the order in which we've performed the steps. The same set of steps is still performed. Are there still an infinite number of balls in the bucket?

Rollback Post to RevisionRollBack

Never attribute to malice what can adequately be explained by incompetence.

You could also look at it as always adding balls and instead of removing them you add "negative balls". This gives the equation infinity - infinity, which does not equal zero (can, but not here).

In what case can it ever equal zero?

Apologies for the rude tone. I don't mean to have a fight, just an intelligent debate. I'll stay civilized.

I remember having a nearly identical debate some years back on another forum. Truth is, I wasn't able to get a satisfactory understanding of how infinite math works until I started taking calculus and number theory in college. It's a difficult concept to properly explain and requires a lot of background, but I think it's something worth learning, even if you have to do it on your own. :smile.gif:

You could say inf - inf = undefined, but that would be silly since you clearly add more balls than you take away, so you have to simplify.

Because depending on how I number those balls and which one of those numbered balls I remove, I can end up with none of them left at 12 despite the fact that at any particular step there's a net increase of 9 balls. In one case, we can definitively say exactly which balls are left in the container: those whose numbers are not multiples of 10.

In the other case, we can definitively say that, for any ball, there is some time at which it is removed, so there can't possibly be any balls left in the container. The reason this is a paradox (of the intuitive variety) is because we only changed how balls were selected and not how many.

9*inf = inf

This is nonsense, infinity is not a number.

Rollback Post to RevisionRollBack

Never attribute to malice what can adequately be explained by incompetence.

Well, at any given time, you can find the number of balls in the bucket using the following:

f(x) = (10-1)x
Where x is the number of times the sequence has repeated.

For every ball in the bucket you've taken out, there are 9 that are still in there. It doesn't make much sense when you try to think about it logically because you're dealing with an infinite sequence (represented in this case by a dichotomy). Infinite sequences do not lend themselves to logical thinking very well. It's best not to think about it.

Infinite sequences do not lend themselves to logical thinking very well.

No, they don't lend themselves to intuitive thinking, which is what people try to apply here (and what you really mean when you say logical).

f(x) = (10-1)x
Where x is the number of times the sequence has repeated.

But this only applies for finite x or the limiting case when x approaches infinity.

Imagine instead if we started with the bucket already filled with all the balls and then removed balls one at a time rather than adding 10 and removing one.

The problem is the treatment of countable infinities and how it rarely results in intuitive results (the result is no less true, however) because people try to apply "rules" they know about the cardinality of finite sets to the cardinalities of infinite sets. A similar problem is Hilbert's paradox of the Grand Hotel which clearly demonstrates how intuition is when it comes to infinite sets.

Rollback Post to RevisionRollBack

Never attribute to malice what can adequately be explained by incompetence.

That's because you clearly don't get the problem. The only difference between the two problems is how the balls we removed are numbered. It's counterintuitive because finite sets don't work this way, but infinite sets can.

Once there is an infinite amount of something, you cannot turn that infinite into anything else, no matter how many balls you removed.

What? How is it you can put an infinite number of balls into a bucket but then not manage to be able to do the opposite?

But in the second scenario you aren't putting infinite balls in and then removing them, you are putting one ball in and then removing it an infinite amount of times

This is what makes it clear that you don't understand the problem, because I have absolutely no idea where you pulled this from. Any particular ball is only ever removed once, at the first step, balls 1-10 are put in the bucket and ball 1 is removed. At the second step, balls 11-20 are put in the bucket and then ball 2 is removed. Besides the fact that it doesn't even make sense to remove a ball that's already been removed, I even set up the second situation I stated outright that it was identical to the first case except for how balls are numbered.

Rollback Post to RevisionRollBack

Never attribute to malice what can adequately be explained by incompetence.

Longer answer: Not all infinities are equal. There are an infinite number of whole numbers (1, 2, 3, etc.) but there are still more fractions (1/2, 2/3, 51/5, etc.)
(if you're putting 10 in then taking 1 out) At 12 o'clock, you would have an infinite number of balls in the bucket, and also an infinite number of balls outside of the bucket. There would be nine times as many balls in the bucket as outside the bucket.

This isn't a calculus problem, it's a set theory problem.

There are an infinite number of whole numbers (1, 2, 3, etc.) but there are still more fractions (1/2, 2/3, 51/5, etc.)

Both the natural numbers and rational numbers are countably infinite. This means that for every natural number, you can assign exactly one rational number and for every natural number you can assign exactly one natural number. Since you can put them in a 1-1 correspondence, they must have the same cardinality. Similarly, the integers, multiples of 10, and primes are also countably infinite.

You're trying to apply a property of finite sets to infinite sets. This particular property doesn't hold for infinite sets; the property being given sets A and B such that B is a strict subset of A and A\B (that's the set difference of A and :cool.gif: is not the empty set, then |A| > |B|. This property does not hold for infinite sets. Hilbert's Grand Hotel demonstrates this fact, which is why I brought it up already.

There would be nine times as many balls in the bucket as outside the bucket.

Integer multiples of infinity don't make the slightest bit of sense, so this is nonsense. Again someone decides to treat infinity like it's a number. It's not a number.

you would have an infinite number of balls in the bucket

Again I must ask: what are the numbers on the balls that you claim are still in the bucket? Two weeks ago I made this point and anyone who has claimed that there are an infinite number of balls in the bucket in the second case has completely ignored it. The key feature of the second case is that every ball that gets put into the bucket later gets removed. If there are balls left in the bucket, which ones are they?

Rollback Post to RevisionRollBack

Never attribute to malice what can adequately be explained by incompetence.

Getting into the mathematics proper, some preliminaries include basic set theory and the concepts of isometry, scissors congruence, and equidecomposability. A chapter "Baby BTs" looks at related paradoxes leading up to the Banach-Tarski Theorem: infinite sets, shifting to infinity and stretching, the Cantor Set, the Vitali Construction, Stewart's "HyperWebster" dictionary, and the Sierpinski-Mazurkiewicz Paradox. And in chapter five comes the actual proof of the Banach-Tarski Theorem.

It turns out that the probability that there are 0 balls in the bucket is 1. Of course, this doesn't actually mean that there *are* 0 balls in the bucket, it just means that in almost all cases there are 0 balls in the bucket (kind of like saying almost all real numbers are irrational).

Rollback Post to RevisionRollBack

Never attribute to malice what can adequately be explained by incompetence.

The Meaning of Life, the Universe, and Everything.

Join Date:
9/30/2010

Posts:
660

Member Details

Also, at 12 AM your body has reached absolute zero resutling in a local disaster. Fortunately you have long since died from overexertion, your body trudging on without 'you'. :-)

How is this a paradox at all? I don't get it...
In case 1, you have infinity balls in the bucket minus the multiples of 10, like you said.
In case 2 you have zero balls in the bucket, because after putting them in you immediately took them back out again. It is one infinity minus itself, therefore equal to zero.

If an infinity were to be subtracted from another infinity that is not itself, then that's when 'confusion' may arise, but confusion is not a paradox.
If a specific infinity is subtracted from itself like "x-x" (where x is any/all values for x) then x-x=0, as we all know.

Now, let's do it again, but change the problem a little bit. Instead of removing balls 10, 20, 30, etc. we will remove balls 1, 2, 3, etc. Everything else remains the same. Now how many balls are in the bucket at 12? This time the answer is none. Why? Because every ball that gets put in also gets removed. Ball n gets removed 2^(1-n) minutes before 12. Since, for every ball we can find a time before 12 at which it is removed, there cannot be any balls in the bucket at 12.

Seems rather paradoxical since at any one step of the process you're increasing the number of balls by 9, so how could you ever empty the bucket? Thus is the nature of infinity. And, just to make this a little more interesting, what if, instead of removing a specific ball you remove a random ball chosen uniformly from the balls that are already in the bucket? How many balls are in the bucket at 12 then?

TL;DR

Warned for pointless spam.At 12 there are an infinite number of balls added, and an infinite number of balls removed, but since the number of balls added and removed are not equal, you cannot say it is zero. comparing one infinity to another by itself is meaningless. Instead, you have to compare the functions that produced those infinities.

Using l'Hôpital's rule, you can figure that as the time approaches 12, the number of balls approaches infinity, not zero.

If you think about it like like this:

n = step = infinite

balls = n*10-1 = infinite

(In actual fact, your problem, even the second one, would never spit out 0 because you never reach infinite therefore you continue to the next step. So other than the empty bucket at 0, the minimum number of balls ever in the bucket at one time is 9)

Of course the example is weird, that's the point, but there's nothing actually mathematically invalid about either case. You can't physically realize this scenario, but that's true of a lot of things in mathematics. Another good example is the Banach-Tarski paradox which shows that you can take a sphere, cut it into a finite number of pieces and then through only translations and rotations assemble those pieces into two spheres identical to the first. The reason that this one isn't physically realizable is because the pieces you chop the sphere into are not measurable (I'm not going to go into measure theory here, since I apparently get a tl;dr after a mere 5 lines, if I could relate it to boobs, I'm sure it'd go over better).

The point is that particular sets of axioms don't always lead to intuitively correct results (but that doesn't mean they're incorrect if they follow consistently from the axioms), a lot of other important results in mathematics rely on these axioms (such as the axiom of choice), so usually mathematicians just accept some of the weirdness.

You could also look at it as always adding balls and instead of removing them you add "negative balls". This gives the equation infinity - infinity, which does not equal zero (can, but not here).

Explanation of a concept, it need not be mathematically accurate, it only needs to represent the concept of their sizes.

I notice how you say you can't do it, then prove it by doing it. For instance, both x and 2x^2 both approach infinity as x does. But at any given finite elapsation (that a word?) of x, the second will always be larger. That is what I am saying above. After any finite iteration in the problem above (the second one from OP) there will always be more balls in the bucket than have been removed. If we iterate towards infinity, the difference between them is infinity, but at the same time an infinite number of balls have been removed, and an infinite number have been added.

I am also flabbergasted that you only look at the symbols and not the concepts.

They're both countable infinities, which is a measure of their cardinality. They are not the same size, since for every element in one, there's exactly one element in the other. All the infinities are countable in this problem and, therefore, they all have the same "size".

If that truly is the case, then you should be able to tell me which balls are in the bucket. In the first case that's easy, the balls that are in the bucket are the balls that aren't a multiple of 10. Which balls are still left in the bucket in the second case?

That's true if you ignore a fundamental structure that we imposed on the problem: the fact that the balls are unique. Try imagining the problem as being broken into two steps. First we fill the bucket with all the balls 10 at a time and we don't remove any. Then, as an entirely separate series of steps, we remove the balls 1 at a time starting with the least numbered ball. The only thing different now is the order in which we've performed the steps. The same set of steps is still performed. Are there still an infinite number of balls in the bucket?

I remember having a nearly identical debate some years back on another forum. Truth is, I wasn't able to get a satisfactory understanding of how infinite math works until I started taking calculus and number theory in college. It's a difficult concept to properly explain and requires a lot of background, but I think it's something worth learning, even if you have to do it on your own. :smile.gif:

Because depending on how I number those balls and which one of those numbered balls I remove, I can end up with none of them left at 12 despite the fact that at any particular step there's a net increase of 9 balls. In one case, we can definitively say exactly which balls are left in the container: those whose numbers are not multiples of 10.

In the other case, we can definitively say that, for

anyball, there is some time at which it is removed, so there can't possibly be any balls left in the container. The reason this is a paradox (of the intuitive variety) is because we only changed how balls were selected and not how many.This is nonsense, infinity is not a number.

f(x) = (10-1)x

Where x is the number of times the sequence has repeated.

For every ball in the bucket you've taken out, there are 9 that are still in there. It doesn't make much sense when you try to think about it logically because you're dealing with an infinite sequence (represented in this case by a dichotomy). Infinite sequences do not lend themselves to logical thinking very well. It's best not to think about it.

No, they don't lend themselves to intuitive thinking, which is what people try to apply here (and what you really mean when you say logical).

But this only applies for finite x or the

limitingcase when x approaches infinity.Imagine instead if we started with the bucket already filled with all the balls and then removed balls one at a time rather than adding 10 and removing one.

The problem is the treatment of countable infinities and how it rarely results in intuitive results (the result is no less true, however) because people try to apply "rules" they know about the cardinality of finite sets to the cardinalities of infinite sets. A similar problem is Hilbert's paradox of the Grand Hotel which clearly demonstrates how intuition is when it comes to infinite sets.

That's because you clearly don't get the problem. The only difference between the two problems is how the balls we removed are numbered. It's counterintuitive because finite sets don't work this way, but infinite sets can.

What? How is it you can put an infinite number of balls into a bucket but then not manage to be able to do the opposite?

This is what makes it clear that you don't understand the problem, because I have absolutely no idea where you pulled this from. Any particular ball is only ever removed once, at the first step, balls 1-10 are put in the bucket and ball 1 is removed. At the second step, balls 11-20 are put in the bucket and then ball 2 is removed. Besides the fact that it doesn't even make sense to remove a ball that's already been removed, I even set up the second situation I stated outright that it was identical to the first case except for how balls are numbered.

Longer answer: Not all infinities are equal. There are an infinite number of whole numbers (1, 2, 3, etc.) but there are still more fractions (1/2, 2/3, 51/5, etc.)

(if you're putting 10 in then taking 1 out) At 12 o'clock, you would have an infinite number of balls in the bucket, and also an infinite number of balls outside of the bucket. There would be nine times as many balls in the bucket as outside the bucket.

Another fun idea to play with: http://en.wikipedia.org/wiki/Hilbert's_ ... rand_Hotel

This is not a spammy link, but rather a handy guide

DISCLAIMER: any diagrams I post should be taken with a grain of salt.

This isn't a calculus problem, it's a set theory problem.

Both the natural numbers and rational numbers are countably infinite. This means that for every natural number, you can assign exactly one rational number and for every natural number you can assign exactly one natural number. Since you can put them in a 1-1 correspondence, they must have the same cardinality. Similarly, the integers, multiples of 10, and primes are also countably infinite.

You're trying to apply a property of finite sets to infinite sets. This particular property doesn't hold for infinite sets; the property being given sets A and B such that B is a strict subset of A and A\B (that's the set difference of A and :cool.gif: is not the empty set, then |A| > |B|. This property does not hold for infinite sets. Hilbert's Grand Hotel demonstrates this fact, which is why I brought it up already.

Integer multiples of infinity don't make the slightest bit of sense, so this is nonsense. Again someone decides to treat infinity like it's a number. It's not a number.

Again I must ask: what are the numbers on the balls that you claim are still in the bucket? Two weeks ago I made this point and anyone who has claimed that there are an infinite number of balls in the bucket in the second case has completely ignored it. The key feature of the second case is that every ball that gets put into the bucket later gets removed. If there are balls left in the bucket, which ones are they?

In case 1, you have infinity balls in the bucket minus the multiples of 10, like you said.

In case 2 you have zero balls in the bucket, because after putting them in you immediately took them back out again. It is one infinity minus itself, therefore equal to zero.

If an infinity were to be subtracted from another infinity that is not itself, then that's when 'confusion' may arise, but confusion is not a paradox.

If a specific infinity is subtracted from itself like "x-x" (where x is any/all values for x) then x-x=0, as we all know.

Where is the paradox?... :mellow.gif: