Paradox!

[Tom Sawyer]

Do you not understand the concept of limit? No one is saying you switch from finite numbers to an infinite number while counting, but the limit of counting numbers is infinite.

I'm saying that your usage of an infinite set to describe the situation is wrong. It has the count going 1+1+1+1...N. Doing that goes on forever but never gets to the end of finite numbers or arrives at infinity. If you represent that with a set which contains infinity, your set does not accurately represent what you are attempting to represent with it.

If you take 3+3+3 and represent it as 3*3 or 3², you get an answer of 9 any way that you do it. If you use some mathematical concept to represent 3+3+3 and the result of using that concept arrives at something other than 9, your concept does not accurately represent 3+3+3. If you do 3*3 and get 10, you're not exposing some kind of paradox inherent in the multiplication tables, you're just doing multiplication wrong. Any way that you correctly represent the equation comes out to the same answer.

It's the same here. If you keep adding one and halving the time, you never arrive at either infinity or noon. If you use a set as a shorthand way to represent that and you arrive at a different answer than if you just do it the long way with sequential addition, your set is wrong and doesn't contain the equation you tried to put into it anymore than a set which was supposed to contain 3+3+3 but ends up equalling 10 isn't actually a set with 3+3+3 in it.

 

[Kharakov]

It's the same here. If you keep adding one and halving the time, you never arrive at either infinity or noon. If you use a set as a shorthand way to represent that and you arrive at a different answer than if you just do it the long way with sequential addition, your set is wrong and doesn't contain the equation you tried to put into it anymore than a set which was supposed to contain 3+3+3 but ends up equalling 10 isn't actually a set with 3+3+3 in it.

Well Tom, not to be a jerk or anything, but.... haha... anyway- what if an observer is traveling at \(\frac{\sqrt{19}}{10}c\)?

As to the actual answer, with a .1 meter exchange distance for the balls (assuming they are not placed all the way into the vase, just moved at .99999999c into and out of the vase by a quick observer (observer as in Fringe before it cheesed out)), you end up with 42*9 balls in the vase before you have to attain superluminal speeds to toss another set of balls in the vise.

Was that a tyop? or a psyop? Not like I planned out my 3423'rd post to have 3*42*3 in it... was it? abc 123 c^2=3^2.... haha.

 

[Tom Sawyer]

It's the same here. If you keep adding one and halving the time, you never arrive at either infinity or noon. If you use a set as a shorthand way to represent that and you arrive at a different answer than if you just do it the long way with sequential addition, your set is wrong and doesn't contain the equation you tried to put into it anymore than a set which was supposed to contain 3+3+3 but ends up equalling 10 isn't actually a set with 3+3+3 in it.

Well Tom, not to be a jerk or anything, but.... haha... anyway- what if an observer is traveling at \(\frac{\sqrt{19}}{10}c\)?

As to the actual answer, with a .1 meter exchange distance for the balls (assuming they are not placed all the way into the vase, just moved at .99999999c into and out of the vase by a quick observer (observer as in Fringe before it cheesed out)), you end up with 2^42*9 balls in the vase before you have to attain superluminal speeds to toss another set of balls in the vise.

Was that a tyop? c42c..

Yes, that was my first response. The time needed to move the balls gets you to noon fairly quickly. It's when you assume infinite speed of changing the balls or just rely on the pure math and not worry about the physical constraints of performing the action that noon never arrives.

 

[Deepak]

Could you enlighten us to let us know when the number of balls in the jar starts decreasing. Please keep in mind that the act of adding and removing the balls is at the same time, according to your paradox. Therefore, you are really only adding 9 balls at a time, not adding one in one step and removing one in another.

The nature of the paradox is that at no point does the number of balls start decreasing. In fact, for all finite numbers it increases. The problem here isn't that finite numbers are being treated as infinite, it's that infinite numbers are being treated as finite.

But, if I compare the set of ball which were added and remain with the set of balls which were added then removed I find that those two sets are indeed the same size.

The statement in the OP specifically treated the steps as distinct. 'place 10 balls in the vase, and remove one ball'. I could just as easily only add 9 and keep a 10th which is not added at each step. If the sizes of the balls which are added and the ones which are withheld are are compared how much bigger is the first than the second?

Well, which interval is it that you feel completes the hour and gets you to noon?

\(\omega\)
{1, 2, 3, ...}
http://mathworld.wolfram.com/OrdinalNumber.html

No one shall expel us from the Paradise that Cantor has created. At least without showing where Cantor went wrong.

 

[bigfield]

That is exactly what convergence means. When we assign a number to the infinite sum, the only possible number to choose is exactly 1. Between any pair of distinct real numbers there is a non-zero distance and convergence means that with enough terms the sum is closer to 1 than any other real number x, so the sum cannot be x. The only logical choice is that the sum is equal to 1.

The sum has no solution, not that it is equal to 1, and therefore it makes no sense to assign any number to the sum. t=12:00pm is an asymptote, which the curve representing the number of balls in the vase never reaches.

The problem with this sort of thinking is that the exact same process is used to show that, for example, the set of even positive integers is in a one-to-one correspondence with the positive integers. If we say we add a 'next' term to either set then we treat the operation atomically. That is to say we add to both in a single step, not to one then the other. The sequencing of this problem sticks in our minds as different because it's specifically asking to do one step then the other.

But once you know that the set of even positive integers is actually in one-to-one correspondence with the positive integers then we have no problem conceiving of them reaching to infinity separately, and if I asked you what the difference between a set the size of the positive even numbers and a set the size of the positive numbers I'd guess that you wouldn't have the same qualms that we can only approximate because we can never actually count that high.

I do have qualms with that, but that is perhaps because my understanding of infinity is limited. To me it doesn't make sense to say two infinite sets are the same size.

Does it make sense to say that two different finite sets are the same size? How do you go about testing that? What happens if we perform the same procedure for infinite sets? The notion of a one-to-one correspondence underlies all of these, and it is just that peoples' intuition break down what we move to infinite sets.

Or perhaps subtracting one infinite set from another isn't applicable to the problem at hand.

The number of balls in the vase isn't (10 + 10 + 10 + 10...) - (1 - 1 - 1 - 1...); it is (10 - 1) + (10 - 1) + (10 - 1) + (10 - 1)... = 9 + 9 + 9 + 9...

 

[bigfield]

The posts here are what the paradox was designed to cause. It is supposed to expose the flawed nature of human intuition when dealing with infinity.

If one process starts listing the numbers 1,2,3,4,... and another process lists the same numbers 1,2,3,4,... except it operates 10 times faster, is there ever a number that the fast process reaches that the slow process does not? What does it even mean to go to infinity 'faster' than something else? Do these infinite processes even make sense as thought experiments?

Are there more even numbers than odd numbers? Are there more numbers divisible by three or numbers not divisible by three? Is the operation 'add 10 and remove 1' underspecified for an infinity of operations? Do we need to specify which ball to remove at each step? Why?

For those interested, this is called the  Ross–Littlewood paradox.

I read the solutions proposed using the numbering systems on the wiki page, and they all seem to be flawed in that they are trying to subtract infinity from infinity to get a non-infinite result.

What if the problem was changed so that you place 1 ball and remove 1 ball at each step? You still subtract infinity from infinity, but how many balls remain?

No, you aren't subtracting infinity from infinity. The number of balls would equal (1 -1) + (1 - 1) + (1 - 1) + (1 - 1)... = 0 + 0 + 0 + 0...

 

[Deepak]

Well Tom, not to be a jerk or anything, but.... haha... anyway- what if an observer is traveling at \(\frac{\sqrt{19}}{10}c\)?

As to the actual answer, with a .1 meter exchange distance for the balls (assuming they are not placed all the way into the vase, just moved at .99999999c into and out of the vase by a quick observer (observer as in Fringe before it cheesed out)), you end up with 42*9 balls in the vase before you have to attain superluminal speeds to toss another set of balls in the vise.

Was that a tyop? or a psyop? Not like I planned out my 3423'rd post to have 3*42*3 in it... was it?

Beero's daemon would have no problem moving at these speeds. More to the point, this isn't the natural science board so I'm not sure if a calculation like this, or one of the total amount of matter in the universe which could be converted to balls and a sufficient vase to envelop them really resolves the paradox in a meaningful way.

 

[beero1000]

Let's try this one more time. The fact that it doesn't make sense with what you think "should" happen is the whole point. It is a paradox because there is nothing logically incorrect with the different conclusions. I can even remove any direct reference to infinity, if it makes it more palatable. If you disagree, which of these premises are wrong?

1. There are no balls in the vase at 11:00am.
2. At any time, if there is a ball in the vase then it was placed in the vase.
3. Every ball placed in the vase is placed in the vase at some time before noon.
4. Every ball placed in the vase at some time before noon has a finite number label.
5. Using the consecutive removal criterion, every ball with a finite number label is removed from the vase at some time before noon.
6. If a ball is removed from the vase at some time before noon then it is not in the vase at noon.

If you accept these premises, then the conclusion that if we use the consecutive removal criterion then there are no balls in the vase at noon is logically valid. The comparison of this result with the intuition that there should be infinitely many balls in the vase shows that infinite numbers do not act like finite numbers in very weird ways.

 

[bigfield]

Let's try this one more time. The fact that it doesn't make sense with what you think "should" happen is the whole point. It is a paradox because there is nothing logically incorrect with the different conclusions. I can even remove any direct reference to infinity, if it makes it more palatable. If you disagree, which of these premises are wrong?

1. There are no balls in the vase at 11:00am.
2. At any time, if there is a ball in the vase then it was placed in the vase.
3. Every ball placed in the vase is placed in the vase at some time before noon.
4. Every ball placed in the vase at some time before noon has a finite number label.
5. Using the consecutive removal criterion, every ball with a finite number label is removed from the vase at some time before noon.
6. If a ball is removed from the vase at some time before noon then it is not in the vase at noon.

If you accept these premises, then the conclusion that there are no balls in the vase at noon is logically valid. The comparison of this result with the intuition that there should be infinitely many balls in the vase shows that infinite numbers do not act like finite numbers in very weird ways.

#5 is the problematic premise. How do you establish that all numbered balls are removed before noon?

 

[beero1000]

Let's try this one more time. The fact that it doesn't make sense with what you think "should" happen is the whole point. It is a paradox because there is nothing logically incorrect with the different conclusions. I can even remove any direct reference to infinity, if it makes it more palatable. If you disagree, which of these premises are wrong?

1. There are no balls in the vase at 11:00am.
2. At any time, if there is a ball in the vase then it was placed in the vase.
3. Every ball placed in the vase is placed in the vase at some time before noon.
4. Every ball placed in the vase at some time before noon has a finite number label.
5. Using the consecutive removal criterion, every ball with a finite number label is removed from the vase at some time before noon.
6. If a ball is removed from the vase at some time before noon then it is not in the vase at noon.

If you accept these premises, then the conclusion that there are no balls in the vase at noon is logically valid. The comparison of this result with the intuition that there should be infinitely many balls in the vase shows that infinite numbers do not act like finite numbers in very weird ways.

#5 is the problematic premise. How do you establish that all numbered balls are removed before noon?

Ball n is removed exactly \(\frac{1}{2^n}\) hours before noon.

 

[Tom Sawyer]

#5 is the problematic premise. How do you establish that all numbered balls are removed before noon?

That's the problem exactly, along with #6. If you arrived at noon, you would have removed all of the numbered balls. You never actually get there, though. Everytime a ball is removed, nine more balls are added. No matter which ball you remove, you always have additional balls in the vase and you never get to noon in order to look back and see which, if any, balls are still there.

 

[bigfield]

Let's try this one more time. The fact that it doesn't make sense with what you think "should" happen is the whole point. It is a paradox because there is nothing logically incorrect with the different conclusions. I can even remove any direct reference to infinity, if it makes it more palatable. If you disagree, which of these premises are wrong?

1. There are no balls in the vase at 11:00am.
2. At any time, if there is a ball in the vase then it was placed in the vase.
3. Every ball placed in the vase is placed in the vase at some time before noon.
4. Every ball placed in the vase at some time before noon has a finite number label.
5. Using the consecutive removal criterion, every ball with a finite number label is removed from the vase at some time before noon.
6. If a ball is removed from the vase at some time before noon then it is not in the vase at noon.

If you accept these premises, then the conclusion that there are no balls in the vase at noon is logically valid. The comparison of this result with the intuition that there should be infinitely many balls in the vase shows that infinite numbers do not act like finite numbers in very weird ways.

#5 is the problematic premise. How do you establish that all numbered balls are removed before noon?

Ball n is removed exactly \(\frac{1}{2^n}\) hours before noon.

And at the same point in time, another ten balls are added and given number labels.

 

[beero1000]

Let's try this one more time. The fact that it doesn't make sense with what you think "should" happen is the whole point. It is a paradox because there is nothing logically incorrect with the different conclusions. I can even remove any direct reference to infinity, if it makes it more palatable. If you disagree, which of these premises are wrong?

1. There are no balls in the vase at 11:00am.
2. At any time, if there is a ball in the vase then it was placed in the vase.
3. Every ball placed in the vase is placed in the vase at some time before noon.
4. Every ball placed in the vase at some time before noon has a finite number label.
5. Using the consecutive removal criterion, every ball with a finite number label is removed from the vase at some time before noon.
6. If a ball is removed from the vase at some time before noon then it is not in the vase at noon.

If you accept these premises, then the conclusion that there are no balls in the vase at noon is logically valid. The comparison of this result with the intuition that there should be infinitely many balls in the vase shows that infinite numbers do not act like finite numbers in very weird ways.

#5 is the problematic premise. How do you establish that all numbered balls are removed before noon?

Ball n is removed exactly \(\frac{1}{2^n}\) hours before noon.

And at the same point in time, another ten balls are added and given number labels.

So what? The premises handle those too.

 

[bigfield]

Let's try this one more time. The fact that it doesn't make sense with what you think "should" happen is the whole point. It is a paradox because there is nothing logically incorrect with the different conclusions. I can even remove any direct reference to infinity, if it makes it more palatable. If you disagree, which of these premises are wrong?

1. There are no balls in the vase at 11:00am.
2. At any time, if there is a ball in the vase then it was placed in the vase.
3. Every ball placed in the vase is placed in the vase at some time before noon.
4. Every ball placed in the vase at some time before noon has a finite number label.
5. Using the consecutive removal criterion, every ball with a finite number label is removed from the vase at some time before noon.
6. If a ball is removed from the vase at some time before noon then it is not in the vase at noon.

If you accept these premises, then the conclusion that there are no balls in the vase at noon is logically valid. The comparison of this result with the intuition that there should be infinitely many balls in the vase shows that infinite numbers do not act like finite numbers in very weird ways.

#5 is the problematic premise. How do you establish that all numbered balls are removed before noon?

Ball n is removed exactly \(\frac{1}{2^n}\) hours before noon.

And at the same point in time, another ten balls are added and given number labels.

So what? The premises handle those too.

No they don't. the premises don't address that at all. The fact that ten balls are added whenever one ball is removed guarantees that not all of the balls can be removed.

 

[Kharakov]

Yes, that was my first response. The time needed to move the balls gets you to noon fairly quickly. It's when you assume infinite speed of changing the balls or just rely on the pure math and not worry about the physical constraints of performing the action that noon never arrives.

That's a fairly limited way of looking at things, isn't it?

 

[Jimmy Higgins]

Every ball placed in the vase at some time before noon has a finite number label.

How can the balls all have finite labels if there are infinite steps?

 

[Kharakov]

Well Tom, not to be a jerk or anything, but.... haha... anyway- what if an observer is traveling at \(\frac{\sqrt{19}}{10}c\)?

As to the actual answer, with a .1 meter exchange distance for the balls (assuming they are not placed all the way into the vase, just moved at .99999999c into and out of the vase by a quick observer (observer as in Fringe before it cheesed out)), you end up with 42*9 balls in the vase before you have to attain superluminal speeds to toss another set of balls in the vise.

Was that a tyop? or a psyop? Not like I planned out my 3423'rd post to have 3*42*3 in it... was it?

Beero's daemon would have no problem moving at these speeds. More to the point, this isn't the natural science board so I'm not sure if a calculation like this, or one of the total amount of matter in the universe which could be converted to balls and a sufficient vase to envelop them really resolves the paradox in a meaningful way.

Yeah, I don't really see a paradox- just an infinite amount of balls in a vise, or was it a vase?

I just like having 3^2 in a post with the implied c^2 and 42... It just happened that way, meaningless coincidence, as usual.

 

[Tom Sawyer]

Yes, that was my first response. The time needed to move the balls gets you to noon fairly quickly. It's when you assume infinite speed of changing the balls or just rely on the pure math and not worry about the physical constraints of performing the action that noon never arrives.

That's a fairly limited way of looking at things, isn't it?

Not really. It's just an inherent limitation of trying to use a word problem to describe a mathematical equation. The analogy can sometimes fall apart.

 

[Kharakov]

Yes, that was my first response. The time needed to move the balls gets you to noon fairly quickly. It's when you assume infinite speed of changing the balls or just rely on the pure math and not worry about the physical constraints of performing the action that noon never arrives.

That's a fairly limited way of looking at things, isn't it?

Not really. It's just an inherent limitation of trying to use a word problem to describe a mathematical equation. The analogy can sometimes fall apart.

Ahhh... ahhh... it was a joke about limits (mathematical limits). Usually, if my words look a bit judgemental, there is a hidden joke.

 

[Tom Sawyer]

Ahhh... ahhh... it was a joke about limits (mathematical limits). Usually, if my words look a bit judgemental, there is a hidden joke.

Oh, I see it now. That's actually really funny if you don't miss it because you're as dense as I am.