Infinity Qs

Ok, there has been much heated debate at my house, at which i am greatly outnumbered, at the answers to these Qs:

1)You have an (countable) infinite number of balls, numbered 1,2,3,..... etc. and a box with infinite size.
At 1/1 minute to noon, you put the balls #1->10 in the box, and take ball #1 out.
At 1/2 minute to noon, you put the balls #11->20 in the box, and take ball #2 out.
At 1/3 minute to noon, you put the balls #21->30 in the box, and take ball #3 out.
You keep doing this, i.e at 1/n minutes to noon, you put in the box balls #10n-9 -> 10n and take out ball #n

The question is: How many balls are in the box at Noon?

N.B. This is thoetical, i.e. it takes no time to put it balls/take balls out of the box.

2)If you want some more to ponder, just think about what would happen if you took the #10n-9 ball out, instead of #n. How many balls wuold be in the box at noon then??

the sum diverges..

Yes, well there are definite answers for both Questions, whether they are integers/infinities etc or not is up to you.

I would say an infinite amount, because the pattern of 1/1 minutes to noon, 1/2 minutes to noon, 1/3 minutes to noon, and so on, never actually reaches noon. It happens an infinite amount of times before noon. Each times, you put in 10 balls and take one out. Doing that infinite amount of times means an infinite amout of balls.

Glad to see someone has replied.

You see, the common sense reply would be infinity, cos surely it is the same as putting in 9 balls each time? Or perhaps the time never gets to noon?

Some things to think about:

1) The time surely has to get to Noon. Otherwise, time will stop, and that doesn't happen too regularly.

2) Assume ball X is in the box at Noon (X is a natural number)
BUT, at 1/X minutes to Noon, this ball was removed.
=> Ball X isn't in the box.
=> there are no balls in the box

Seemingly, this last one suggests 0 balls in the box, yet the 'logic' suggests infinite (9 balls put in each time)

So, which is true??

If you are doing this "in theory" then technically you put in infinite number of balls and you take out infinite number, so that's infinity - infinity. Or you can argue, 10*infinity-infinity. You do the math on that and decide.

From my point of view, if you take into the account all the laws of physics, and assume that there really is a mechanism that can put the balls in and take them out in virtually no time, eventually the speed at which this needs to be done will reach light speed, and assuming that you can't go above that, you will no longer be able to put in balls before time reaches noon. Also, if you say that it takes no time to put balls in or take them out then you are saying that motion is done in a period of 0 seconds which is impossible.

So looking at this problem from theoretical point of view, there are a number of solutions, which are all true. But if looking at this problem from realistic point of view, there are many limits on it and that would create a definite answer. You just need to take all the variables into account (i.e. speed of the mechanism that puts balls in/takes them out)

As already posted by Kedaman, the following series diverges.

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1/1 + 1/2 + 1/3 + 1/4 . . . 1/11 . . . 1/n

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The terms up to 1/11 add up to more than 1, so it looks like you would have 90 to 99 balls in the box at noon.

For a convergent series, there would be an unbounded number of stages, and the problem would be more interesting. An infinite number is likely to be the correct answer.

At each stage the number of balls increases by 9. For any finite number of stages, the number of balls in the box grows without bound. I find it hard to believe that something magic happens at or near infinity and the numberof balls decreases.

Perhaps at inifinity they all vanish in a cloud of Pixie Dust, otherwise you have a lot of balls in that box.

At noon, the box will be empty.

This is because, if you compare the cardinality of the set of balls put in, with the set of balls taken out, they are the same (in the same way that the set of integers has the same cardinality as the set of squares).

SimonM: You could be correct, but I do not think so. I agree that the cardinality of the two sets is the same, indicating that their difference is zero. I do not like this conclusion, but then much of Cantor’s work is counterintuitive and logically correct.

I am not sure that your analysis corresponds to the description of the problem.

If I told you that I was adding 9 balls to the box for each term in the series, you would surely agree that there would be a lot of balls in the box whenever I decided to count them.

The problem describes putting in ten and removing one at each stage. This is equivalent to increasing the count by nine at each stage. Arriving at a zero count by subtracting two unbounded sets does not seem correct. Why should the following two sums be different?

SumA = 9 + 9 + 9 + 9 + 9 + . . . .

SumB = (10 -1 ) + (10 -1 ) + (10 -1 ) + (10 -1 ) + . . . .

It does not seem correct to evaluate the second sum by rearranging the terms to create two unbounded sets.

Yeah, this is what i was hoping would have happened. An intriguing question.

MXK, you are right, in a 'real' sense this cannot be done, as the time intervals grow too small. This is why i put the 'Theoretical' part in.
However, your response that there are a number of solutions if it is theoretical is confusing.

Guv, it seems i may not have explained the time thing well. The first ball is taken out at at 1/1 min. to noon.
The next ball is taken out at 1/2 min to noon, 1/2 min after the 1st. the next ball is taken out at 1/3 min TO noon, 1/6 min after the 2nd etc. (1/6 = 1/2 - 1/3)

If you add all these times up, you get a telescoping sum:
(1 - 1/2)
+(1/2 - 1/3)
+(1/3 - 1/4)
+(1/4 - 1/5) .....

You see they all cancel out leaving you with 1 minute, which is the time to noon, as wanted.

Oh, and the mention of (10-1)*infinity = 9 is kinda weird
If u look at the first Q, (10-1)*infinity = 10*infinity - 1*infinity = 0 (seemingly, if 0 is the answer)

However, looking at the 2nd part (10k+1 th ball removed) this seems to be (10-1)*infinity = 9 *infinity = infinity

it seems to be, when does (10-1) <> 9??

How you doing Guv? :)

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The problem describes putting in ten and removing one at each stage. This is equivalent to increasing the count by nine at each stage. Arriving at a zero count by subtracting two unbounded sets does not seem correct. Why should the following two sums be different?

SumA = 9 + 9 + 9 + 9 + 9 + . . . .

SumB = (10 -1 ) + (10 -1 ) + (10 -1 ) + (10 -1 ) + . . . .

It does not seem correct to evaluate the second sum by rearranging the terms to create two unbounded sets.

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It is a strange conundrum...

Because, you wouldn't end up with zero balls if you merely added balls 1-9 and then left out the tenth, then added balls 11-19 and left out 20 etc...

I guess this is just one of those problems that appears to have a different solution depending on how you approach it.

I maintain that the sum grows without bound.

Rearranging an infinite series to create two series is not a valid operation. It leads to paradoxes. I remember pseudo proofs resulting in paradoxical results due to such rearrangements.

Consider the following infinite series, outside the context of this thread.

9 + 9 + 9 . . .

It is clearly unbounded and it is clearly the same as the following unbounded series.

[10 - 1] + [10 -1] + [10 -1] . . .

I do not think that rearranging to create the following is valid.

[10 + 10 + 10 . . .] - [1 + 1 + 1 . . .]

The latter becomes the difference of two unbounded series, creating a bit of confusion, and suggesting that the sum might be zero, which is a silly result considering the original series.

Proposed Question for "Who wants to be a Millionaire?"

You have taken a 50-50, and only have answers a and c left

For the million dollars, what is (10-1)*infinity:
Is it:
a) 10*infinity - infinity = 0
c) 9*infinity = infinity

(eerie music plays in the background, your palms grow sweaty, what will your answer be...or do you take the $500000?)

Guv

I know what you're saying but, can you deny these facts:

The set of all the balls put in (ignoring those taken out) will be a set with the same cardinality as the set of integers.

The set of all the balls removed will be a set with the same cardinality as the set of integers.

The cardinality of a set is the number of elements it contains.

Therefore, there must be zero balls left at the end of the operation.

hey guys,

i haven't studied infinity in earnest but i think im going to stick with guv on this one.
because we keep increasing n by 1 every iteration, 1/n will grow closer and closer to zero but never really get there. and as a consequence of this the number of balls in the box can be compared to the number of balls taken out in that they are both infinities. but if we could quantify infinity we'd find that the infinity of the number of balls within the bow would be greater than the infinity of the number of balls taken out thus rendering the proposition infinity of balls in box - infinity of balls outside box = 0 invalid. but what this would assert is :

infinity of balls in box > infinity of balls taken out

TheAlchemist

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because we keep increasing n by 1 every iteration, 1/n will grow closer and closer to zero but never really get there.

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Another one in infinity denial. :rolleyes:

The process was structured in such a way that an actual inifinity of steps would be completed in one minute.

It's similar to the following sequence. These are intervals of time (seconds):

1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2

Each time interval is half of the one that preceeded it. The sum of the series will be equal to 2 seconds. The series is infinitely long but shows how an infinite series can be summed to provide a finite total.

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infinity of balls in box > infinity of balls taken out

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Actually, it can be proven that they are equal infinities. The are both infinite sets with the same cardinality.

Cantor showed that the size of infinte sets can be compared.

If the elements of two sets can be mapped on a 1-1 correspondence, they must have an equal number of elements.

The set of balls taken out can be mapped with a 1-1 correspondence with the set of balls taken out and therefore they have the same cardinality.

I think you folks are claiming that many (perhaps all) divergent series sum to zero. To repeat previous posts with a slight variation.

N + N + N + . . . = [N - 1] + [N - 1] + [N - 1] + . . .

[N + N + N + . . .] - [1 + 1 + 1 + . . .], which is equivalent to the right side of the previous expression.

The latter is the difference between two sets with same cardinality. Hence the latter expression is zero.

Therefore an obviously divergent series sums to zero.

Do you really believe this?

I accept that the cardinality of various sets are the same even though this result is often counterintuitive.

I do not accept operations which decompose an infinite series into two transfinite sets. Some seemingly ordinary operations are just not allowed when working with infinite sets.

To repeat my argument. Why are the following two series not equivalent?

9 + 9 + 9 + 9 + . . .

[10 - 1] + [10 - 1] + [10 -1] + . . .

If they are equivalent, both grow without bound, and neither sums to zero.

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The set of balls taken out can be mapped with a 1-1 correspondence with the set of balls taken out and therefore they have the same cardinality.

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simonm - i dont understand the above could you explain?

i also don't know what the cardinality of an infinite set means could someone explain that too.
i have manged to find out how an infinite set can sum up to a limit. from this wouldn't it mean that the time offset from twelve would eventually add up to 1(?).
lim as n->infinity (1/n1 +1/n2 ........) = 1 minute

but i still dont understand why the infinity of balls in the box are equal to the infinity of balls taken out. isn't the former greater by a factor of 9(since for every one ball taken out,nine are added)

Alchemist: I am very uncomfortable about what are called transfinite numbers, due to the counterintuitive nature of the subject. A man named Cantor formalized the subject prior to 1900.

Cardinality is a fancy term for how many members a set has. Cardinal numbers specify how many. Ordinal numbers specify position in a sequence. There are fifty people in line at the ticket office. I am the fifth in the line. Fifty is cardinal, while fifth is ordinal.

Cantor analyzed infinite sets and coined the term transfinite numbers. He started by pointing out that the cardinality of sets could be determined by trying to match members.

If the members of two sets can be put into one-to-one correspondence, they have the same number of members or the same cardinality. If the matching process results in one set having some unmatched members, it has more members and the greater cardinality. Very simple and intuitively easy so far.

Then he pointed out that you could match all the even integers with all the integers. There are as many even integers as there are total integers. Try it. Start writing a list of all integers. In a parallel list write a list of all the even integers. You get the following pairs: (1,2), (2,4), (3,6), (4,8) . . . For every even integer, there is a corresponding integer in the other list, and vice versa.

He used this property as the fundamental definition of a set with a transfinite number of members: Such a set could be put into a one-to-one correspondence with a subset of itself. He went on to prove some interesting theorems, including the following.

• The set of all rational numbers has the same cardinality as the set of all integers. There is a systematic way of matching the members of these two sets.
• There are more real numbers than integers. The number of real numbers is called the Power of the Continuum. It is the number of points on a line.
• The number of points on a short line has the same cardinality as the number of points on an infinitely long line.
• The number of points in a 2D or 3D space is the same as the number of points on a line, or 1D space.
• If you make a set of all the subsets of a set, it has a larger cardinality than the original set.

All very counterintuitive. He called the cardinality of the set of all integers Aleph-0 (using the first letter of the Hebrew alphabet). Of course this is the same as the cardinality of various other sets. The set of all subsets of the set of integers he said had cardinality Aleph-1. Similarly, you can define Aleph-2, Aleph-3, et cetera.

I think he proved that the Aleph’s were all the transfinite sets there could be, by proving that there could be no transfinite number between Aleph-n & Aleph-(n+1).

I think there is an unproven conjecture about the Power of the Continuum being Aleph-1.

Does the above help or confuse you? Most of it confuses me. I agreed with all the logic when I read about transfinite sets, but find the conclusions counterintuitive. Emotionally, I think there are more rational numbers than integers, but intellectually I agree with the proof that both sets have the same cardinality.

Guv

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To repeat my argument. Why are the following two series not equivalent?

9 + 9 + 9 + 9 + . . .

[10 - 1] + [10 - 1] + [10 -1] + . . .

If they are equivalent, both grow without bound, and neither sums to zero.

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I cannot argue with your logic and I agree that, when you put it like this, the assertion that there will remain zero balls left in the box at noon seems ridiculous.

The only thing I can suggest is that perhaps you are losing something in your abstraction. This is because the exact pattern the balls are added and removed affects the end result. i.e. If I add balls 1-10 and then remove ball 10, add 11-20 and remove 20, etc... I will not end up with zero balls at noon.

Both these examples correspond to your abstraction so perhaps your abstraction is missing something?

:D ;) :cool: THANKS A LOT GUV

wow that really helped! simonm, i think guv is right in saying that the series

[10-1]+[10-1]+[10-1]+........

cannot be decomposed into:

[10+10+10+10.....]-[1+1+1+1.....]

because we need to compute whats inside the brackets first(brackets have a higher precedence than addition)
thus giving

9+9+9+9+9.......

whose sum is infinity

TheAlchemist

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i think guv is right in saying that the series

[10-1]+[10-1]+[10-1]+........

cannot be decomposed into:

[10+10+10+10.....]-[1+1+1+1.....]

because we need to compute whats inside the brackets first(brackets have a higher precedence than addition)
thus giving

9+9+9+9+9.......

whose sum is infinity

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Perhaps that's the problem. The series does not need to be re-arranged into:

[10+10+10+10.....]-[1+1+1+1.....]

because that is how it should be represented to start with.

It seems to me that this whole question deals with our view of inifinity. How many balls are in a box after an infinite number of repetitions? That question can only have a definitive answer if infinity is finite.

Yes, you can ask how many balls are in the box at noon, but that is taking you out of the theoretical framework of the question. We aren't capable of taking no time to add balls to the box, so in reality, this situation can't happen. If we say that it does happen, then we are moving away from reality in the direction of being able to infinitely divide the minute before noon. Once we move away from reality in this direction, then we never will reach noon, and the number of balls in the box will never be evaluated.

Some are saying there will be 0 balls, while others are saying there will be an infinite number. I would suggest that both are correct...simultaneously. There are an infinite number of real numbers, but if you were to try to write them out, you would miss at least an infinite times the number of numbers that you wrote. Why? Because we are dealing with infinity as being "a really large number" when in fact it is just a concept. You can't "write out" all the numbers, except in theory. That theory is a concept, a conceptual interpretation of the idea of infinity. The concept is inadequate to encompass the reality of infinity, and the errors in the concept are sufficient to allow these paradoxes to exist.

Consider the balls in the box. We are saying effectively: "This can never really happen, but consider if it did." In other words, "pretend that this were possible". Once we make up a world to pose a question, we can make up an answer that satisfies us. If the answer you find satisfying is 0, then so be it. Likewise, if the answer is infinity, then so be it. You are the observer, you decide the outcome. If you were to try to test this, you would run into the physical issues of trying to do what cannot be done. If you tried to come as close as you could to the theoretical ideal, you would still be working with finite values, and not infinite values. In that world, there would be a specific number of balls in the box, and the number would be greater than 0.

Infinity is a concept, not a value. Using it as a specific value can cause weird issues like this.

I understand what u are saying, and most of it is correct (like, infinity is a concept, and that you never get past 0% when writing all the numbers)

However, some of the things i disagree with.

1) A question can have a definitive answer if infinte IS infinite. Take the sum of an infinite series. This has a definite value, yet there are truly INFINITE terms. (Infact, if there were a finite number of terms, the value would be less/more than when there are infinite)

2) "we are moving away from reality in the direction of being able to infinitely divide the minute before noon"
I hate to point out, but not many people realise that you CAN infinitely divide up any minute, and still get to the end. You see, in the time it takes you to read this post, there must have been a half-way point. And 1/4 of the way through. And 1/8, and 1/16 etc. Any fraction X between 0 and 100 (of which there are an infinite number of the same magnitude as we are talking about), then there was a time which you were X% through reading this. Yet, in REALITY, you have reached the end. :D

I agree with the first point, but largely because time is not an issue in evaluating an infinite series.

As to the second point, what does that say? This is like Zeno's paradox. If you shoot an arrow at somebody running away, the arrow will travel half the initial distance, but in that time, the person has moved a little bit further on. In less time, the arrow will travel half the remaining distance, but the person will have moved just a little further, etc. The arrow can never reach the person because the person will always move a small amount forward in the time it takes the arrow to cover x% of the distance where x<100.

At some point, the observation itself influences the evaluation of the event. You may observe the first balls going into the box and the first ball coming out of the box, but the time for each event is decreasing. As you approach noon, an infinity of instances stacks up with all balls coming out of the box and an infinity more going into the box. I would suggest that in the instant before noon, both are true, an infinity entering the box, and an intinity exiting the box.

You can evaluate some infinite series, but not all of them. Some, like this box, have only conceptual answers.

where would you, of all people, find that many balls?

Shaggy hiker: Discussing infinity does not give us license to ignore logic and semantics. What we write should still be reasonable.

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How many balls are in a box after an infinite number of repetitions? That question can only have a definitive answer if infinity is finite.

Some are saying there will be 0 balls, while others are saying there will be an infinite number. I would suggest that both are correct...simultaneously.

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Bolding in the above by me. When phoney Eastern mystic gurus were in fashion, people were enthralled by phrases like The sound of one hand clapping. Such phrases should be considered nonsense or worse yet, semantically meaningless. They are not profound. Neither are obviously paradoxical statements.

Others: Infinite series which converge are not so bad, but infinity, transfinite numbers, et cetera are a mess to deal with. It is mathematical logic with no real world models to guide us. Infinity, singularities, sets with an unbounded number of members, et cetera do not exist in what we call the real world. They do exist in the mindscape of mathematics and can be dealt with there using logic.

To me, this thread has created confusion by an unnecessarily complicated description of an infinite process. Broken down, the process describes nine balls being put into the box at each stage. Ergo, there is no way for there to be zero balls in that box at nay stage.

I think the finite time aspect of the problem is a red herring which merely makes the problem more confusing. Would the problem be different if the schedule was once per second forever?

A different process could be described with one demon putting infinitely many balls into the box using one schedule, while another demon using a different schedule takes one ball out infinitely many times.

The latter process presents a problem. The result is (infinity - infinity), which might not be analyzable. The result might be zero or infinity or some inbetween value. Two bright people could probably present arguments for two different results and confuse the hell out of me. There might be a different result if the box starts with some balls in it, so the demon removing balls can start first. There is sure to be controversy if the loader demon has a finite schedule, but works infinitely fast, while the removal demon works at finite speed, but has infinite time.

BTW: It is improper to ask questions like What is 0 / 0 or Infinity / infinity? without some context.

x / sin(x) approaches one as x approaches zero. Other variations on this theme approach infinity or various finite values other than zero. Similarly, infinity/infinity or (infinity - infinity) can have a logically consistent value depending on the context.

You object to the statement that a system can be in two states simultaneously? That isn't goofy eastern mysticism, but modern quantum theory.

I like your suggestion that we consider this operation once per second forever. That will reduce some confusion, perhaps. However, we are asking what the state is at the end of this series. The series never ends, so there is no state at the end of an unending series. If you want to consider it at the hypothetical end, then both possibilities are possible, and this becomes a similar issue to Shroedingers cat (is that spelled right?). We can't know the exact state until we observe it, we never will observe it, therefore it is simultaneously in both states. If you find that to be logically ridiculous, read the book Entanglements for a fine description of very real paradoxical seeming consequences of quantum mechanics.

As for your objection to my first statement, I agree that I was playing with semantics, but the point is valid. We are pretending that there will be an end to infinity; an end to forever, and asking what happens at that point. However, that point, if there is a point, is finite. Evaluate the function when forever ends? You will never evaluate. Infinity is infinity, and any finite approximation alters the original question. For any finite approximation of infinity, there will be an enormous number of balls in the box, never 0. The argument stating that there will be 0 balls only holds if you are really talking about infinity, and not just a "really really long time".

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"However, we are asking what the state is at the end of this series"
and:
"We are pretending that there will be an end to infinity"

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1) we are asking what the state is at the end of an infinite number of time differences, which in all are the same as 1 minite.
=> The infinite series has and end. Not an end to the series, but the end of time at the SUM of the series. And infinite series can have sums, so i'm not sure what the problem is.

2) We are not pretending that there will be an end to infinity. We are saying an infinite number of things can happen in a finite amount of time (like dividing a minute into parts of a minute)

Also: The idea of the archery thing helps.
Consider the archer and the "victim". The victim stands x meters away from the archer. The arrow moves to the archer at 10 m/s, the victim runs from the arrow at 1m/s (all in 1D)
=> Net result = arrow gains on archer at 9 m/s

=> if X is a real (positive) number, the the time for the arrow to hit the victim can be worker out. (although there are infinite intervals)

HOWEVER, if x = infinite, the arrow NEVER hits the victim. Every step the arrow is no closer.
Somehow, this has relevance, though don't ask how. Possibly cos if you do stuff at anything OTHER than infinity, you get a 'normal' value, yet once you take it to infinity, you don't.

Shaggy Hiker: Quantum theory is relevant to this thread? That is why you said

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Some are saying there will be 0 balls, while others are saying there will be an infinite number. I would suggest that both are correct...simultaneously.

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Quantum theory is even more confusing than I thought.

The concept that a situation can be in two states simultaneously is relevant to this thread. After all, most examples I know of use macroscopic events to demonstrate quantum principles. Some people feel that an unobserved system that can be in multiple states with similar probabilities will be in all of those states simultaneously until they are observed. Only then will the exact state be resolved. This appears to be true for light, in that it will behave as a wave or a particle depending on how it is being measured.

I would say that this situation fits pretty well. People have made reasonable arguments for two different and mutually exclusive results. Since nobody has come up with any solid argument saying why either result MUST be wrong (except by asserting that the alternative result MUST be right), then it appears that either could be right. I would say that both are right as long as the number of repetitions are infinite.

Of course, if the number of repetitions are not infinite, then there will be a readily calculable number of balls in the box, and the number will be greater than 0 if the number of iterations is also greater than 0.

If the number of iterations are infinite, as is postulated, then the number of balls in the box can take on either of two values: 0 or infinity. The system doesn't have to fall into either state until it is measured. Since it never can be measured, it remains in both states.

This isn't quantum mechanics, but the idea of duality such as this arose from quantum theory, and can be demonstrated most easily in quantum studies, which is why I referenced that.

I don't feel that there is an answer along the lines of: "There are X balls in the box." I feel that the answer is that there are either infinite or 0. That isn't very satisfying, since we don't really like definitive questions with indeterminable answers, but we ought to get used to it. Quircky little things like this are a fact of life. Even with language these things arise. For instance, there are sentences that can be spoken easily, but cannot be written correctly. If we can come up with this type of paradox in our own language, we can come up with even better ones in math.

I just realised a weird scenario which is exactly the same to this:

Consider an infinitely long stretch of road (the 'box')
There are two cars side by side, A and B (the actions of moving 'balls')
Car A travels at 10m/o (10 meters per 'operation' = +10 balls)
Car B travels at 1m/o (1 meter per 'operation' = -1 ball)
How far apart will they be, after an infinite number of 'operations'?

Well, this is exactly the same as the question being asked. If you label each meter from the starting point 1,2,3 etc, then at the first operation, car A has 'added' the 1st 10, while car B has 'taken' #1, etc which happens in the question.

Now, the argument some (incl. me) are saying is 'for every meter car A travels, car B travels as well' => they are next to eachother.
Just something to think about.

It's a good analogy. I don't think we can know the answer, and I'm still advocating an indeterminate "both" solution.

heres a cool link on the Heinsburg Uncertainity Theorem,the basis of modern quantum phyics.

Quantum Theory

Alchemist: The link you provided is interesting. It seems to make the Uncertainty Principle the result of some arbitrary decisions by Heisenberg and Born, as well as suggesting that there is some dubious algebraic manipulation involved. I do not think this was the intent of the article, but it is my interpretation.

It also implies that Bohr did not agree with them. Perhaps Bohr did not agree at first, but I am sure that he agreed very soon after it was published.

However shaky its origins, the Uncertainty Principle, it is a firmly held concept now. The Bose-Einstein condensate is one of many incredible experiments that verify the UP. The BEC results alone are compelling evidence in favor of the UP, especially since the properties were predicted 20-30 years or more prior to the experiment being performed.

Shaggy Hiker: You are still advocating zero and infinity simultaneously? Be serious. There might be some argument for undecidable, but those two possibilities are mutually exclusive.

The issue here is very clear. Either the problem describes the following series or it does not.

10 - 1 + 10 - 1 + 10 - 1 . . .

Grouping (not rearranging) results in the following.

(10 -1) + (10 -1) + (10 -1) . . . This is clearly 9 + 9 + 9 + 9 . . .

The above series is clearly infinite.

Another interpretation of the problem is the following.

(10 + 10 + 10 . . .) - (1 + 1 + 1 + 1 . . .)

In mathematical slang, the latter interpretation is (infinity - infinity), which at least requires further analysis. Such analysis might conclude zero or indeterminant. The cardinality of those two infinite sets is the same Aleph-0 described by Cantor. If their difference is defined, I suspect that it is zero.

However, converting the first series into the second is not an allowable operation. The result depends on your interpretation of the original problem. If you want to argue for zero, accept the second interpretation. If you want to argue for infinity, accept the first interpretation, which seems like the correct one to me. You cannot argue for both results.

The two interpretations clearly lead to two mutually exclusive results. Claiming that both interpretations are correct is silly. The resulting conclusion that two mutually exclusive results are simultaneous possible is also silly. Reductio ad absurdum proofs use silly results to prove valid theorems. Your conclusion that both results are simultaneously true indicates that your logic is faulty or one of your premises is wrong.

If claim that the problem statement is ambiguous, then you can draw no conclusions, but there is no basis for claiming that zero and infinity are simultaneous vvalid.

Rearranging the terms of a divergent infinite series to create two infinite series is an unallowable operation known to result in paradoxical conclusions. Rearranging the terms of a convergent series to create two divergent series is also a no no. For example the following series converges.

1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 . . .

Rearranging it into the difference of two divergent series is not allowed and can result in silly conclusions.

There are some situations in mathematics for which one of two solutions is possible with no way to choose. For example: The sum of the following infinite series is either one or zero, but not infinity.

1 - 1 + 1 - 1 + 1 . . .

Such situations are usually pretty obvious. The subject of this thread is not such a situation.

Trying to quantify infinite....

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(countable) infinite number of balls

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You realize that's a contradiction right?

:rolleyes:

You cannot have an infinite amount of balls, so the question is irrelevent.

Instead, quanitfy the number of balls. Say... 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 999999999999

Hope this helps....

:D

And the answer is infinite.

Because for every iteration there would be 9 more balls, so an infinite amount of iterations yeilds: 9 * infinity, and welp, there ya go. 9 * infinity = infinity.

Guv, Einstein argued stongly against the implications of quantum uncertainty, but current studies suggest he was wrong. He really wanted a deterministic result to the problem, but the results don't appear to be determined until they are measured, and the measurement itself influences the answer. I am saying that this is a similar situation. The system doesn't take on either state until it has been measured, and isn't one or the other until that measurement has been made. Further, I suggest that the measurement itself, impossible though it is, determines the result.

I don't agree that everything must be this or that, even in math. In some situations, it can be both until measured. Wishing it away is very logically satisfying, but it isn't right. Fortunately for you, if you choose to do so, you are in the best of company. Einstein himself, argued stongly for a definitive answer, but the weight of evidence is against him by now.

Two mutually exclusive conditions simultaneously occuring? Not something we like to think about in our every day life. We would prefer to believe that if we just knew a little more about a system immediately before we observed it, then the results we observed would be a logical extension of that. However, that is a mistake that our mind is creating. Even college level physics labs can perform simple experiments with light demonstrating a system simultaneously existing in two mutually exclusive states. Is that silly? No, unless you want to believe it is mass halucination. What does it suggest? To me it suggests that we are making a fundamental mistake in logic. After all, logic, of which we are so fond, is not itself a complete system. You complain that something is illogical, even irrational, but it isn't hard to find examples in science these days where the illogical is easily proven. Do those studies not exist? Do their results never happen? How do you square illogic with logic. Some would discount it all as silly or trivial, but it is generally the flaws in a system that lead us to greater understanding.

Shaggy Hiker: You do not seem to understand transfinite arithmetic. You do not seem to be aware that Quantum weirdness is applicable to the Quantum world, not the classical world and certainly not the world of mathematical logic. Quantum Theory relates to physics, not mathematics, although mathematics is used in the discipline.

The world of mathematical logic one does not need to make a measurement to come to a conclusion. If logic implies a result, it is valid without measurement.

Perhaps You are a precocious, but naive, 12 year old who has learned some Quantum theory jargon without understanding it. Perhaps you are older and took some courses that confused you to the point that you think Quantum uncertainty applies universally.

Perhaps you are just too stubborn to give up a lost argument.

I repeat my previous point of view. You really have three choices.

• Claim that the original post was semantically ambiguous. In this case there can be no futher analysis. I hope you understand the implicaitons of the term semantically ambiguous. From what you have posted so far, the concept might be beyond your intellectual capabilities.
• Claim that the original post describes the series (10 -1) + (10 -1) + (10 -1) . . . In this case the number of balls grows without bound.
• Claim that the original post describes the diference between two infinite series. In this case the number of balls might be indeterminant or might be zero . For all I know, it might be infinite in this case. I do not remember enough about transfinite arithmetic to decide this case.

There is no alternative that allows the claim that the number of balls is simultaneously zero and infinity. That claim is nonsense.

1) Semantically Ambiguous?? :p

3) In this case the number of balls might be indeterminant, or might be zero. - I think we are saying its zero.

However, for those who think its infinite, how about this induction:

n=k=1
Ball 1 is taken out at action #1
=> True for n=k=1

n=k+1
Ball n is taken out 1/n(n-1) minutes after ball (n-1)=ball k is taken out
as ball k is taken out (proved earlier) => ball n is taken out.

=> For all n in the set of integers (which all the ball numbers are), ball ne is taken out, and so is not in the box at the end.

It is an interesting question though, isn't it :D ;)

hey sql_lall,
i really didn,t understand what you were trying to say with the ns and ks, maybe you should try put in some commas and all.
shaggy hiker: i think you're approach is very interesting,its certainly given me a new perspective on the topic but
i think GUV is right in stressing the mathematical aspect of this thread and the importance of this in the way we choose to approach the topic. Heres a little thought to hold!

Mathematicians can say whatever they like but physicists must be at least partially sane.
--J Willard Gibbs(professor of mathematics at the yale university a long long time ago)

by the way guv,you're interpretation of the thread was correct, the guy who wrote it intended exactly that.

Guy who wrote it being me, or the person I got the problem from? Just for interest sake, i got the problem from a book i got recently, called "Dr. Riemann's Zeros", by Karl Sabbagh. There is a bit at the end (Toolkit 3) about infinite series, and this problem is at the start. In the book it says it comes from "Littlewood's Miscellany", by J.E. Littlewood.

Oh, and incase you are wondering, the book gives the answer as zero. However, on the FIRST PAGE they list the primes INCLUDING 1!!! :eek: So i'm not sure how acurate the maths in the book is :p

BTW: waht i was trying to say that was for every ball that is taken out, the ball with the value 1 greater than it was taken out, => all balls were taken out. (Induction)

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However, on the FIRST PAGE they list the primes INCLUDING 1!!!

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What's wrong with that? :confused:

Quantum physics only works in the quantum world? The last I heard, every partical up to and including a human would demonstrate the same wave/particale duality as light if it were to travel slowly enough. The quantum world is not some esoteric theoretical world divorced from so-called reality. The principles work in the macroscopic as well as atomic level.

So you want to believe that math must come to a conclusion? Must there be an answer? Therefore one of the arguments for infinity or zero is wrong. I disagree.

As for my background, I am 35, and have been a scientist my entire life. I discovered my first mathematical theorem back in high school, and have used original lines of thought in designing a few mathematically oriented programs to locate solutions to chaotic problems. In the course of my career, I have encountered many people who reject certain solutions because they are too complex, and indefinite. Try talking chaos theory with biologists (or economists, if they'll listen) and you'll see what I mean. We have caused more harm than good in this country (USA for those who aren't sure) by looking for a single concrete answer and rejecting the premise that there isn't one.

Guv, this really isn't the forum, due to the time delay, for any kind of discussion on whether there must be an answer to this question, but I urge you to look into this more closely. I get the feeling from your posts that you are another person who wants the world to be black and white. I too, would prefer this be so, but I have found more interesting and practical solutions to problems by considering what would happen if the coin landed on edge.

And if you'd just prefer a flame war, I can do that too, but I'd rather not.

Hey, chill, chill.. Lets just talk about the maths.


Anyway, simonm, 1 is NOT a prime!!! it has only 1 distinct positive factor, not the 2 that are needed for primality. Oh, that wasn't the only problem in the book, there were a few more, i'll see if i can find them again :D

Shaggy Hiker

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Quantum physics only works in the quantum world? The last I heard, every partical up to and including a human would demonstrate the same wave/particale duality as light if it were to travel slowly enough. The quantum world is not some esoteric theoretical world divorced from so-called reality. The principles work in the macroscopic as well as atomic level.

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It is my understanding also that quantum effects to not manifest themselves on the macro scale (or at least they become extremely unlikely).

The rules that appear to govern the macro world of classical mechanics some how emerge from the micro world of quantum uncertainty.

I think there is a danger of applying quantum results to situations that do not warrent it and I believe you are doing so now.

sql_lall

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Anyway, simonm, 1 is NOT a prime!!! it has only 1 distinct positive factor, not the 2 that are needed for primality.

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Well, as I understood it, a prime number must only be divisible by one and itself. Since 1 is divisible by 1 and itself (even if they happen to be the same number) it would qualify as a prime number.

Perhaps I have been unclear, and I'm sorry that Guv ticked me off. I don't think that exchange could have occurred except through a faceless medium like this.

I feel that both sides, zero and infinity, have made compelling arguments for their interpretation. However, I have seen no mathematical argument against either side. The arguments against each side have been logical ones along the lines of "This shows that the answer must be x, and therefore, other answers can't be right." However, both sides are making that argument, and doing so in a manner sufficient to prevent invalidation on a mathematical basis. I say that because there hasn't been such an invalidation yet.

In the early days of math, or so we have been told, there was no 0. That was added, then negative numbers, fractions, decimals, etc. At some point people found that there were non-repeating decimals (e.g. pi) for which there is a name that I forget. However there was still a gap.

At some point, people began accepting calculations using unreal numbers (i) for which there really isn't any logical real-world representation. Surely there were those who felt that using the square root of a negative number in your calculations was absurd.

During the Age of Reason, mathematicians were looking for a "complete" system of mathematics. A complete system was one with the power to define and identify all valid theorems within the system. Godel proved that wasn't possible. No system can be created that can derive all theorems, and once you get close to such a system you can derive at least one theorem that is both true and not true, a paradox.

In math, as in physics, we can find paradoxical results once the system becomes sufficiently close to completeness. The current thread is a similar case, since solid arguments can be made to "prove" two mutually exclusive results.

I am disturbed by this incompleteness we are seeing in some systems. Usually, when we find flaws in the patterns coming from theoretical science, the reason is that our comprehension is inadequate. Being a biologist, I don't think humans are particularly intelligent, since our actions aren't substantially different from other animals, but in this one area, our intelligence is our asset. We must keep moving forward in knowledge. A series of problems of this sort, arising as they do in different fields, suggests that current models can be improved upon. Numbers are only a model of the world around us. We have already expanded that model to inculde unreal numbers, perhaps we need to, again, expand the model.

We know of equations (quadratics) which have two algebraic solutions that are both valid, but are not at all contradictory. Perhaps we will eventually come up with a further set of math to cover equations that have two contradictory solutions. This thread is an example of such a problem. As the number of iterations approaches infinity, the number of balls in the box approaches infinity, but at infinity itself, the number of balls in the box can be either infinite or zero, as many have demonstrated. Why can't there be a further set of numbers beyond unreal.

Perhaps the inconceivable numbers.

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Originally posted by sql_lall
Glad to see someone has replied.

You see, the common sense reply would be infinity, cos surely it is the same as putting in 9 balls each time? Or perhaps the time never gets to noon?

Some things to think about:

1) The time surely has to get to Noon. Otherwise, time will stop, and that doesn't happen too regularly.

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Obviously, the closer you get to noon, the faster you have to move, in order to get the ever increasing number of balls into this infinitely voluminous box. So, Naturally, you end up approaching the speed of light, where you will start to observe some relativistic effects on time. Therefore, can you actually guarantee that, for the person adding and subtracting the balls, that he will ever get to noon?

Sounds like a greek tragedy to me.
:)

I wish those who are discussing this issue would pick from the following alternatives or propose some fourth possibility.

• The original post is semantically ambiguous. In this case there can be no further analysis. I hope you understand the implications of the term semantically ambiguous.
• The post describes the series (10 -1) + (10 -1) + (10 -1) . . . In this case the number of balls grows without bound.
• The original post describes the difference between two infinite series. I think this alternative leads to zero.

To me it is very clear that the answer depends on which of the above interpretations one chooses.

I would not argue with any valid analysis which starts by mentioning which of the above interpretations is considered correct. A valid analysis can only be refuted by claiming that it misinterprets the problem description.

The finite time frame seems to be obfuscation. The process describes an unbounded number of operations, although the time schedule was not precisely described in the original post. Sql_Lall clarified in a later post, resulting in a schedule with an unbounded number of operations. Briefly, that post said the following (using numbered balls).

Put in #1- #10 & take out #1
Put in #11 - #20 & take out #2
Put in #21 - #30 & take out # 3
. . . .

Simonm: Please note below, the difference of two transfinites with same cardinality need not be zero. You interpret the second part of the operation as removing balls #1 to #n, which is clearly the removal of an unbounded number of balls. After reconsidering a bit, I think this interpretation is correct or at least describes the intent of the author. I change my vote to agree with Simonm: Zero balls are in the box when the process is complete (if ever). The result is the difference of two transfinites with the same cardinality.

It is strange, but consider the following change in the description.

Put in #1 - #10 & take out #1
Put in #11 - #20 & take out #11
Put in #21 - #30 & take out # 21
. . . .

The latter describes putting in ten and taking out one, or adding nine at each step. Clearly, balls #2 - #9, #12 - #20, et cetera are never removed. The result is a lot of balls in the box after a lot of steps. A transfinite numbers of balls are removed, but a transfinite number remain.

Both descriptions specify putting in ten and taking out one at each step, but are subtly different. Transfinite arithmetic is counterintuitive.

If you describe two independent processes.

• The former could be described as put in ten balls at each step for an unbounded number of steps, followed by the removal of all the balls one at a time.
• The latter could be described as put in ten balls at each step for an unbounded number of steps, followed by removal of every tenth ball starting with #1. This is the removal of an infinite subset, but not all the balls. Clearly, an infinite subset remains. Following are never removed: #2 - #10, #12 - #20, et cetera.

I would like to see a description that leads to both zero and an infinite number simultaneously as the answer to the above, which I still claim is a silly and/or meaningless statement.

Shaggy Hiker: I do not view everything as black or white, although the world of mathematics seems to have definite answers to many questions. I certainly do not view Quantum objects as well defined until measured.

However, I do not think it is proper to say that light is both a wave and a particle until some experiment results in measuring some wave or particle property. I seem to remember descriptions using words like Limbo State, implying neither wave nor particle until measurements are made. Similarly for the properties of other quantum objects.

Perhaps your claim for simultaneously existing paradoxical values is not even valid in the Quantum world. Do you have any good citations for such simultaneity? I seem to remember incorrect conclusions resulting from making assumptions about the state of an unmeasured quantum property.

Sorry Guv, I don't have any citations handy, I might be able to dig some up in a day or so, but I guess it doesn't really matter.

I prefered your earlier stance that the answer was not 0, but agree that there is a good argument for 0. I'd rather not vote on this, but would prefer that there were an answer. I suppose I choose the semantically ambiguous option, though semantically ambivalent seems more correct. I don't think the ambiguity lies in the semantics, but that may just be more antics with semantics.

Firstly, i am not sure what part of the Questin is ambiguous.

However, i can say that you never reach the speed of light. NotLKH, we have assumed that putting balls in/out of the box takes no time at all.

I think the problem comes with treating the balls as objects, not every ball as '1'.

You see, if we treat every ball as '1', and adding/subtracting to box as +/-, then sure, the sum ends up as (10-1)+(10-1)+(10-1) etc. = 9+9+9... = infinity

However, the balls are objects, you can't take any 1, you have to take a certain 1.
I.e. the sum then becomes:
(1₁+1₂+1₃+1₄+1₅+1₆+1₇+1₈+1₉+1₁₀ - 1₁)
+ (1₁₁+1₁₂+1₁₃+1₁₄+1₁₅+1₁₆+1₁₇+1₁₈+1₁₉+1₂₀ - 1₂)

This is a telescoping sum, which cancels out to 0.
I.e. if the balls ar treated as a value, "1" then you get infinity. If you treat them as objects (i.e. strings: "ball1", "ball2" etc) then they cancel out to 0.

This is what i don't agree with in Guv's (10-1)+(10-1)+... case, as you are really taking an arbitrary "1", from a 10 which the 1 isn't in. Sounds weird, i know.

Sql_Lall: Check my most recent post, which agrees with you and SimonM.

In particular note that the finite time schedule is unnecessary confusion.

ive browsed this site of and on ever since i started teaching myself VB, but it wasnt till i saw the truely in-depth discussion of this thread that i felt i had something to contribute.:D

DISCLAIMER: im not a mathematician, i just like playing around with it. im not a physicist, i just find it interesting

as i understand it, any hypothetical universe actually DOES exist, even if it is only a mathematical construct. what i mean is: any hypothetical set laws is just as valid as the set under which we live (so long as it does not violate itself). so this experiment could actually take place once a set of laws is in place. the true outcome (or measurement) really depends on these laws. so in effect Shaggy Hiker was right in saying, "We can't know the exact state until we observe it..." but not because of the uncertanty princible or because of a physicist's cat. we will know the outcome simply because we set it to be true by the predefined laws.

this i think is the only instance of symantical ambiguity, but only because we cannot seem to agree apon what the laws ARE. the only constant it seems is time. since the problem statement suggested that we could *expand:eek:* one minute to perform an infinite number of operations while maintaining that it is a finite amount time, i put forth my (tenative) conclusion:

the answer seems to 0. since each ball that ever enters the box eventually leaves it, it shouldnt matter how quickly the box is filled or emptied, so long as the entire process is completed before noon.:D

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or because of a physicist's cat

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hehe :D

If it were a dog, many more people would have liked Shroedinger (sp?)
However, as it is a cat, no-one really cares. Even if it were dead, it would still come back as the ghost of uncertainty past.

good point,

Actually, I once read a book about quatum theory in which the protagonist meets Shroediger's cat. The cat relates that as the experiment was taking place, he escaped the box while the physists weren't looking. when they opened the box to discover that there was no cat inside, they were unable to make the necessary measurement. thus, the cat was instantly doomed to wander the earth in constant flux between life and death.