Paradox!

[Speakpigeon]

As Juma pointed out, we are dealing with a series, not a sequence, which is a sum of the terms of a sequence. A convergent infinite series does not get closer and closer to a value, it is exactly equal to the value. Wikipedia  Series_(mathematics)#Convergent_series gives a simple proof for (almost) this very series. Note that it is an exact equality, not a limit. (The Wiki series starts with 1 instead of 1/2 so it is equal to 2 and not 1. Just subtract off the initial 1 to get our series which is equal to 1.) Noon will be reached in exactly 1 hour.

No, we're not dealing with a sum. The process in this case is counting (v. adding balls in the original OP process). Each counting takes place at a particular time, each before 12:00, as specified, and therefore nothing is specified for 12:00. More importantly, there is no "last count" being done at any time because the series is infinite. So, when "we" arrive at 12:00, as I guess "we" must, "we" don't have any last count available.

Summation only applies to our view of time. The successive periods of time we have spent must be added, so we get to 12:00, but what we are at 12:00 depends on the laws of physics. What the process you specified becomes at 12:00 is not specified and cannot be inferred from physical laws since we're not in any realistic situation since the process itself, as specified, is not possible in the real world.

PS. I'll look up your link but not now, I'm low on sugar.
EB

 

[Kharakov]

Zenos paradox is solved by fact that sums with infinite number of terms can be finite. Has nothing to do with reality.

Exactly what I said.

Thanks for clarifying my thoughts.
EB

A finite number of thoughts can be used to understand that an infinite number of terms can be summed up to a finite amount. Nice that there are finite bounds to infinity, ehh? Booooiinnnggg....

 

[Artemus]

As Juma pointed out, we are dealing with a series, not a sequence, which is a sum of the terms of a sequence. A convergent infinite series does not get closer and closer to a value, it is exactly equal to the value. Wikipedia  Series_(mathematics)#Convergent_series gives a simple proof for (almost) this very series. Note that it is an exact equality, not a limit. (The Wiki series starts with 1 instead of 1/2 so it is equal to 2 and not 1. Just subtract off the initial 1 to get our series which is equal to 1.) Noon will be reached in exactly 1 hour.

No, we're not dealing with a sum. The process in this case is counting (v. adding balls in the original OP process). Each counting takes place at a particular time, each before 12:00, as specified, and therefore nothing is specified for 12:00. More importantly, there is no "last count" being done at any time because the series is infinite. So, when "we" arrive at 12:00, as I guess "we" must, "we" don't have any last count available.

Summation only applies to our view of time. The successive periods of time we have spent must be added, so we get to 12:00, but what we are at 12:00 depends on the laws of physics. What the process you specified becomes at 12:00 is not specified and cannot be inferred from physical laws since we're not in any realistic situation since the process itself, as specified, is not possible in the real world.

PS. I'll look up your link but not now, I'm low on sugar.
EB

Alright, you've agreed that noon will be reached, but still believe that there needs to be a "last step" for the OP problem to make mathematical (but not physical) sense. I doubt I can convince you that there doesn't need to be one, but if you really want to understand try reading about  telescoping series and how changing the order of terms are included (under "a pitfall" in the wiki link) can affect the value to which it converges (or if it does at all). That is the significance of the labeling scheme for removing the balls that beero1000 was describing.

 

[steve_bnk]

Time does not have any independent existence. Meters and seconds are arbitrary 'rulers' used to quantify change in observable reality.

Asking whether time is continuous or not is the same as asking if the meter is continuous. In the end the limit is quantum scale particles. A meter stick is comprised of particles and is not infinitely divisible.. Whether that means space itself, without defining what that may mean, is another question.

That would be Zeno's Paradox framed in modern science, can motion take on any infinitely small steps. I would think basic quantum mechanics wold say no. If energy transfer is quantized and motion takes energy then Zeno's Paradox may be answerable.

The current definition of the second is based on counting discreet events.

http://en.wikipedia.org/wiki/Second

The master second is then divided down electronically to smaller steps. A stop watch that electronically counts in fractious of a second is traceable to the international second.

In practical terms the tie resolution is limited by how high a clock frequency can be generated, and the speed f light along with quantum considerations.

A simple interval counter. I have an electronic circuit that is gated for 1 second and counts the number of transitions on a signal. A signal frequency of 1000 Hz for 1 second would ideally show a count of 1000, and a time resolution of 0.001 second. The smallest time interval that can be resolved is limited by the speed of light within the electronic components. Electronics is also limited by the electron as as a discrete particle, and quantized energy transfer.

So time as is it scientifically defined is not infinitely divisible, at least as I see it from basic principles. Asking a question like whether 'time itself' is continuous to me has no meaning.

Whether or not something is treated as continuous or discrete depends on the model and situation.

Theoretically electric current is quantized by the electron which is supported by experiment. Current is charge, number of electrons, divided by time passing thorough a surface, dq/dt.

In circuit models and calculations in most cases we treat current as continuous taking o any value limited by the number of digits used, with the measured resolution set by the instrumentation.

Similarly consider a ball in motion. At the Newtonian scale we consider the ball taking taking on any calculated increment velocity, limited by instrumentation resolution.


As I remember it as you go from quantum to Newtonian the density of states becomes so fine that they appear continuous. Similar to the transition of state in a transistor. Unless you are designing materials and transistors, we treat the transition as continuous while there are actually a finite number of discrete steps.

All our scientific definitions are ultimately not continuous.

 

[Speakpigeon]

Alright, you've agreed that noon will be reached

Ah, my point was what exactly would reach noon!

I agreed that we could assume that the observers, or people doing the counting, sort of exist in time and therefore will reach noon. That's to accommodate your specification that time should be treated in a realistic way. But I would still maintain that we don't know that it is true that time is really continuous. More to the point, as some other posters say, even if time is continuous, what matters is whether physical process are themselves continuous in time or instead work by quantised jumps, including quantised jumps in time, in which case all infinite sequences are non-realistic and the logic of series may provide an inappropriate framework.

Alright, you've agreed that noon will be reached, but still believe that there needs to be a "last step" for the OP problem to make mathematical (but not physical) sense.

I had already agreed that 11.999... is equal to noon:

To go back to the point argue by other posters, personally, I would accept that 11.9999999999... = 12. However, there is not even an nMax such that: f(nMax) = 11.9999999999...

This is the same as saying that certain series do converge. If the sequence is: f(0)=11 and f = f(n-1) + 9/(10^n), then the series is 11.9999999999... = 12.

Further, the number of balls in the vase in the OP problem goes up to infinity (i.e. there is no upper bound). And I still don't see why removing 1 ball for each time we add 10 could possibly end up with the vase being empty at 12:00am. None of you seemed prepared to articulate this result properly.

Me, I would put it tentatively like this: EACH ball which is added at some point in time before noon will be removed at some point in time before noon.

Now, to infer from this that ALL balls will have been removed at 12:00 is a mistake. Yes, EACH ball will be removed before 12:00, but there is no reason to infer that ALL balls will have been removed at 12:00. Sure, if you could prove this, you'd have a paradox, in the sense that, intuitively, most people would say that we end up with an infinite amount of balls and you would prove them wrong somehow. But the proof offered isn't acceptable, not as presented at any rate.

Further, the demonstration given by beero1000 shows that we can so "prove" that the number of balls left at noon can be chosen to be 42 or any number we wish to obtain according to the order in which we remove the balls. I would say this isn't a vote of confidence in the sanity of the demonstration.

I also note that according to the Wiki page you referenced, mathematicians since the 17th century have endlessly and violently argued about this and at least until "rigor" (rigor mortis?) finally set in, I believe at the end of the 19th century, which left a great deal of time indeed to argue.

I wouldn't dispute mathematical proofs but I think there is room to argue whenever a mathematical problem is couched in ordinary language, as was the case here. The way I characterise the number of balls left is different from that of beero1000. I certainly remain to be convinced.
EB

 

[Bomb#20]

A meter stick is comprised of particles and is not infinitely divisible..

We don't know that. A lot of things once thought to be elementary turned out to be divisible. Maybe there are sub-quarks, and maybe there are sub-sub-quarks, and maybe...

Theoretically electric current is quantized by the electron which is supported by experiment. Current is charge, number of electrons, divided by time passing thorough a surface, dq/dt.

No, theoretically electric charge is quantized by the electron which is supported by experiment. That's no reason to think current is quantized. Just because you can't have pi electrons doesn't imply you can't have pi electrons per second passing through a surface. You'd just have to have 1 electron passing through it every 1/pi seconds.

So in order to show current is quantized, you'll need an experiment that supports time being quantized. That's a problem. Suppose there's a chronon, a smallest unit of time. Suppose event Y happens one chronon after event X. Now watch X and Y from a different inertial frame of reference moving relative to them at relativistic speed. From that point of view, Y happens less than one chronon after X.

So if you want to show current is quantized, first you'll need an experiment that shows relativity is wrong.

 

[Speakpigeon]

Alright, you've agreed that noon will be reached, but still believe that there needs to be a "last step" for the OP problem to make mathematical (but not physical) sense. I doubt I can convince you that there doesn't need to be one, but if you really want to understand try reading about  telescoping series and how changing the order of terms are included (under "a pitfall" in the wiki link) can affect the value to which it converges (or if it does at all). That is the significance of the labeling scheme for removing the balls that beero1000 was describing.

Something else. In the case of the number of balls (original OP problem), if we follow Beero1000 reasoning, we can add each time 10 properly labelled balls and then remove one according to a scheme determining how many balls are left in the vase at 12:00am. If the reasoning is accepted and the result is considered as the number of balls left at 12:00am, then the reasoning is self-contradictory. This is not so much making the case for a proper paradox, i.e. a reasoning that shows common opinion to be somehow misguided, but a contradiction in terms, from which therefore you cannot prove anything, let alone that common opinion is wrong. Rather, as I broadly said early on, you are putting forward a reasoning that doesn't make sense and challenging people to find where it goes wrong.

Except that you seem to believe that the reasoning doesn't err and does prove something. Which I don't know what it is exactly.

As presented by the OP, each step of the process adds 9 balls in the vase. If you consider the series made up of the total number of balls left in the vase after each step, i.e. 9 + 9 + 9 + 9... ad infinitum, the series has no finite limit and can't possibly be 42 or anything finite. Doesn't that shows that there is something wrong in the reasoning based on certain labelling schemes?

There's indeed a warning in the Wiki page, that you have pointed out yourself, about the pitfall of certain telescoping schemes. The telescoping series corresponding to Beero1000's "paradox" would be 10 - 1 + 10 - 1 + 10 - 1 etc. ad infinitum. The most straightforward telescoping scheme is to change this into 9 + 9 + 9 + 9 etc. ad infinitum, which trivially does not converge at all. So, as far as I can tell, Beero1000 didn't avoid the pitfall and selected the wrong kind of telescoping scheme. In particular, the Wiki page says that the series should be absolutely convergent, which I understand invalidates Beero1000 telescoping scheme.
EB

 

[Artemus]

Something else. In the case of the number of balls (original OP problem), if we follow Beero1000 reasoning, we can add each time 10 properly labelled balls and then remove one according to a scheme determining how many balls are left in the vase at 12:00am. If the reasoning is accepted and the result is considered as the number of balls left at 12:00am, then the reasoning is self-contradictory. This is not so much making the case for a proper paradox, i.e. a reasoning that shows common opinion to be somehow misguided, but a contradiction in terms, from which therefore you cannot prove anything, let alone that common opinion is wrong. Rather, as I broadly said early on, you are putting forward a reasoning that doesn't make sense and challenging people to find where it goes wrong.

Except that you seem to believe that the reasoning doesn't err and does prove something. Which I don't know what it is exactly.

As presented by the OP, each step of the process adds 9 balls in the vase. If you consider the series made up of the total number of balls left in the vase after each step, i.e. 9 + 9 + 9 + 9... ad infinitum, the series has no finite limit and can't possibly be 42 or anything finite. Doesn't that shows that there is something wrong in the reasoning based on certain labelling schemes?

There's indeed a warning in the Wiki page, that you have pointed out yourself, about the pitfall of certain telescoping schemes. The telescoping series corresponding to Beero1000's "paradox" would be 10 - 1 + 10 - 1 + 10 - 1 etc. ad infinitum. The most straightforward telescoping scheme is to change this into 9 + 9 + 9 + 9 etc. ad infinitum, which trivially does not converge at all. So, as far as I can tell, Beero1000 didn't avoid the pitfall and selected the wrong kind of telescoping scheme. In particular, the Wiki page says that the series should be absolutely convergent, which I understand invalidates Beero1000 telescoping scheme.
EB

It was going to drop this, but...

The pitfall is not that the answers are incorrect, the pitfall is that changing the ordering will lead to a different answer, which was my point and the gist of the OP problem. You are demanding that the series be interpreted as

\(S = \sum_{n=1}^\infty (10 - 1),\)

meaning that one of the balls removed is one of the balls that was placed. That series is divergent. But that is but one of an infinite (cardinality \(\aleph_0\)) interpretations. If you remove the first ball placed the first time and the second ball placed the second time etc. the series is

\(S = \sum_{n=1}^\infty(1 - 1) = 0.\)

which converges to zero. The key is that the cardinality of the two infinities are the same so a one-to-one correspondence can be obtained (mathematically termed  bijection). If you decide on the other hand to remove balls sequentially beginning with the fourth added you would get

\(S = 1 + 1 + 1 + \sum_{n = 4}^\infty (1 - 1) = 3.\)

Depending on the removal ordering chosen, you can get a mathematically valid answer equal to any non-negative integer, or an infinity of cardinality \(\aleph_0\).

ETA: The wiki does not say that the series must be absolutely convergence, it says that the series must be absolutely convergent for any reordering to give the same answer.

 

[Kharakov]

This is why hyperreals are useful. 9H

 

[steve_bnk]

We don't know that. A lot of things once thought to be elementary turned out to be divisible. Maybe there are sub-quarks, and maybe there are sub-sub-quarks, and maybe...

Theoretically electric current is quantized by the electron which is supported by experiment. Current is charge, number of electrons, divided by time passing thorough a surface, dq/dt.

No, theoretically electric charge is quantized by the electron which is supported by experiment. That's no reason to think current is quantized. Just because you can't have pi electrons doesn't imply you can't have pi electrons per second passing through a surface. You'd just have to have 1 electron passing through it every 1/pi seconds.

So in order to show current is quantized, you'll need an experiment that supports time being quantized. That's a problem. Suppose there's a chronon, a smallest unit of time. Suppose event Y happens one chronon after event X. Now watch X and Y from a different inertial frame of reference moving relative to them at relativistic speed. From that point of view, Y happens less than one chronon after X.

So if you want to show current is quantized, first you'll need an experiment that shows relativity is wrong.

Anything is possible. The center of the galaxy might be cheddar cheese.

Google 'Millikan oil drop experiments', one of the original demonstrations. In the SI system the electron is assigned a unitary value, the unit charge.

Particle physics has a strong empirical foundation n its predictive power. If you want you want to make a claim for a continuous current you will have to replace modern physics

in the early 1900s there were competing theories, one being likened to a continuous fluid. The electron won out.


Water is comprised of discrete molecules. In a water pipe we use mass flow rate, kilograms per second through a surface.

In the early 1900s xray diffraction of crystalline materials showed discrete nodes(atoms) in regular structures. Small nodes separated by relatively vast inter-atomic distances The idea of a solid material was shattered, and actually caused some popular consternation.

Again, the meter and second are arbitrary measures of observed reality, they are not reality itself whatever that may be.

A thought experiment. Take a meter stick and repeatedly bisect into smaller divisions. At some point you reach the atom quantized condition as a limit.

You can hypothesize a deeper continuous reality but to me that is problematic. How would you cut a continuous material? My sense is a continuous reality is precluded by potential singularities.

 

[Artemus]

OK, I tried to hammer this out very quickly and a couple things don't make sense. As it is too late to edit, I've put corrections in italics...

I was going to drop this, but...


The pitfall is not that the answers are incorrect, the pitfall is that changing the ordering will lead to a different valid answer, which was my point and the gist of the OP problem. You are demanding that the series be interpreted as

\(S = \sum_{n=1}^\infty (10 - 1),\)

meaning that the ball removed in a step is one of the ten balls that was most recently placed. That series is divergent. But that is but one of an infinite (cardinality \(\aleph_0\)) interpretations. If you remove the first ball that was placed the first time a ball is removed and the second ball that was placed the second time a ball is removed etc. the series is

\(S = \sum_{n=1}^\infty(1 - 1) = 0.\)

which converges to zero. The key is that the cardinality of the two infinities are the same so a one-to-one correspondence can be obtained (mathematically termed  bijection). If you decide on the other hand to remove balls sequentially beginning with the fourth added you would get

\(S = 1 + 1 + 1 + \sum_{n = 4}^\infty (1 - 1) = 3.\)

Depending on the removal ordering chosen, you can get a mathematically valid answer equal to any non-negative integer, or an infinity of cardinality \(\aleph_0\).

ETA: The wiki does not say that the series must be absolutely convergent, it says that the series must be absolutely convergent for any reordering to give the same answer.

Next time I'll wait until I'm not so rushed.

 

[Bomb#20]

So if you want to show current is quantized, first you'll need an experiment that shows relativity is wrong.

Anything is possible. The center of the galaxy might be cheddar cheese.

Google 'Millikan oil drop experiments', one of the original demonstrations. In the SI system the electron is assigned a unitary value, the unit charge.

Particle physics has a strong empirical foundation n its predictive power. If you want you want to make a claim for a continuous current you will have to replace modern physics

in the early 1900s there were competing theories, one being likened to a continuous fluid. The electron won out.

Oh for gods' sake! Can you seriously not tell the difference between the concept "charge is quantized" and the concept "current is quantized"? What you say here amounts to accusing me of denying that charge is quantized. Of course charge is quantized. Duh! No, I am not proposing to replace modern physics. You are the one who is proposing to replace modern physics -- modern physics does not say current is quantized. So do not lecture me about oil drop experiments. Those experiments show charge is quantized.

If you're so sure modern physics says current is quantized, please tell us how many amperes the minimum quantum of current is.

 

[steve_bnk]

'....No, theoretically electric charge is quantized by the electron which is supported by experiment. That's no reason to think current is quantized...'

Quantized by the electron but not really quantized???

okey dokey....like I said propose an alternative to modern physics.

Again read up on Milikan's oil drop experiments and then get back to me. Charge quantization is not just theoretical. I used to have an original copy of his book The Electron with all his data. He is not as well known but was up therewith Einstein and the rest. You can Google the ampere. It is defined as flow rate of electrons per unit time.

http://ffden-2.phys.uaf.edu/212_fall2003.web.dir/ryan_mcallister/slide3.htm


http://www.physics.nist.gov/cuu/Units/current.html

'...The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per meter of length. …'

Current is flow rate of charge, and charge quantized as the electron. Current can only change in increments of the electron. This is what Millikan found, charge can only be transferred in integer multiples, inferring a single discrete unitary charge.

i = current
q = harge
t = time
i = dq/dt.

1 amp = 1 coulomb per second.

http://en.wikipedia.org/wiki/Coulomb#Definitionhttp://en.wikipedia.org/wiki/Coulomb#Definition


'... In the SI system, the coulomb is defined in terms of the ampere and second: 1 C = 1 A × 1 s.[3] The second is defined in terms of a frequency which is naturally emitted by caesium atoms.[4] The ampere is defined using Ampère's force law;[5] the definition relies in part on the mass of the international prototype kilogram, a metal cylinder housed in France.[6] In practice, the watt balance is used to measure amperes with the highest possible accuracy.[6]

Since the charge of one electron is known to be about 1.60217657×10−19 coulombs, a coulomb can also be considered to be the charge of roughly 6.241509324×1018 electrons (or protons), the reciprocal of 1.60217657×10−19....'

The actual value assigned to the electron has been rationalized so all current theory based on SI units are consistent.

We usually treat current as continuous because the typical instruments we use do not get close to the sensitivity of a unitary charge. There are solid state devices and sensors which respond to small numbers of charges.

Special instruments will read to femto amps. Routine instruments resolve mili amps to micro amps.

Steve's the name and current is my game...heeheeeheeeheeeheee...

This would be a derail over to science.

 

[Speakpigeon]

OK, I tried to hammer this out very quickly and a couple things don't make sense. As it is too late to edit, I've put corrections in italics...

I was going to drop this, but...


The pitfall is not that the answers are incorrect, the pitfall is that changing the ordering will lead to a different valid answer, which was my point and the gist of the OP problem. You are demanding that the series be interpreted as

\(S = \sum_{n=1}^\infty (10 - 1),\)

meaning that the ball removed in a step is one of the ten balls that was most recently placed. That series is divergent. But that is but one of an infinite (cardinality \(\aleph_0\)) interpretations. If you remove the first ball that was placed the first time a ball is removed and the second ball that was placed the second time a ball is removed etc. the series is

\(S = \sum_{n=1}^\infty(1 - 1) = 0.\)

which converges to zero. The key is that the cardinality of the two infinities are the same so a one-to-one correspondence can be obtained (mathematically termed  bijection). If you decide on the other hand to remove balls sequentially beginning with the fourth added you would get

\(S = 1 + 1 + 1 + \sum_{n = 4}^\infty (1 - 1) = 3.\)

Depending on the removal ordering chosen, you can get a mathematically valid answer equal to any non-negative integer, or an infinity of cardinality \(\aleph_0\).

ETA: The wiki does not say that the series must be absolutely convergent, it says that the series must be absolutely convergent for any reordering to give the same answer.

Next time I'll wait until I'm not so rushed.

I still have a couple of objections to what you say here.

First, in Wiki on Grandi’s series, it says (my emphasis):

It can be shown that it is not valid to perform many seemingly innocuous operations on a series, such as reordering individual terms, unless the series is absolutely convergent. Otherwise these operations can alter the result of summation.

Now, the series

\(0 = \sum_{n=1}^\infty(10 - 1)\)

is not absolutely convergent so I don’t see how it’s valid to do the telescoping you’re doing and I don’t see it as correct.

Second, in Wiki on Grandi’s series, it says

Although telescoping can be a useful technique, there are pitfalls to watch out for:

\(0 = \sum_{n=1}^\infty 0 = \sum_{n=1}^\infty(1 - 1) = 1 + \sum_{n=1}^\infty(-1 + 1) = 1\)

is not correct because this regrouping of terms is invalid unless the individual terms converge to 0; see Grandi's series. The way to avoid this error is to find the sum of the first N terms first and then take the limit as N approaches infinity:

But this telescoping is the one you are doing.

Further, I don’t see that

\(0 = \sum_{n=1}^\infty(1 - 1)\)

can represent at all the problem as it was specified. Sure,

\(\sum_{n=1}^\infty(1 - 1) = 0\)

is true but it doesn’t represent the problem as specified. We were supposed to add 10 balls and then remove one at each step. I will grant you that we can choose the ball we remove, in the way you suggested, by taking out the one ball which has been there the longest so to speak but that’s still adding 10 balls each time we remove one. Take the process one step at a time, much as I think it is suggested above with partial sums, and you have obviously successive steps with partial sums each 9 balls more than the previous one. The limit of these partial sums is clearly the infinite, not 42 or 3. So how you can accept that the process specified can somehow produce 42 balls at twelve o’clock is beyond me.
Plus, according to you, the process also results in 10 balls, 1239870 balls, 47 balls etc in the vase, all at the same time, 12:00am. That’s not logical. Like

\(0 = \sum_{n=1}^\infty 0 = \sum_{n=1}^\infty(1 - 1) = 1 + \sum_{n=1}^\infty(-1 + 1) = 1\)

says that 0 = 1, we would end up saying that 10 = 1239870 = 47 = 42 = 3, yes?

Here, it seems relevant too to consider this definition given by Wiki on Grandi’s series:

In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists.

So, the equivalence between the sum of an infinite series and the limit of the sequence of its partial sums is not demonstrated as the logical consequence of some first principles, it is just given as a definition. You accept this definition if you want to but it is just as legitimate to refuse it.

Finally, a series is defined as “the sum of a sequentially ordered finite or infinite set of terms”. In other words, each ordering defines a different series. The point then is which series, and therefore which ordering, it is appropriate to use for which problem. That’s also where we diverge to infinity in our respective opinions.
EB

 

[Artemus]

I'll just make one more post and then leave this for good. Think of it in terms of set theory. The set of all balls removed from the bucket when noon is reached is infinite with a cardinality of \(\aleph_0\) (a countable infinity). The following three subsets of the balls added are also infinite sets with a cardinality of \(\aleph_0\):

1. The set of all balls added to the bucket.
2. The set of all balls added to the bucket from the fourth ball on.
3. The set of every tenth ball added.

A bijection can be established between the set of balls removed with any of the three subsets listed by specifying the order in which the balls are removed. If the bijection is with subset 1, all balls are removed and the remaining subset is empty. There are no balls left at noon. If the bijection is established with subset 2, the remaing subset is the first three balls added so three remain. If the bijection is established with subset 3, the remaining set consists of balls 1 through 9, 11 through 19, 21 through 29 ... This is an infinite set of cardinality \(\aleph_0\). You can get any non-negative integer or a countably infinite remainder depending on the order of removal choosen.

 

[Speakpigeon]

Sure, as I already said, I don’t dispute the math and I accept the notion it’s possible to match N and Q one to one (bijection). But the debate is not about the math. The terms into which the “paradox” has been couched (balls, vase, and a repetitive, mechanical albeit infinite process of adding and removing the balls from the vase) suggest that the world considered is a non-realistic one but also one which is physical, in the sense of possessing physical laws, not quite our own laws but laws nonetheless. So we certainly expect some sort of deterministic answer.
In the Zeno paradox, nobody is really suggesting that Achilles won’t ever catch up with the turtle, or will and will not catch up. No, in fact we all know he will. The paradox is in fact a challenge to articulate why he will. When it comes to the balls-&-vase “paradox”, however, you seem to be saying that the potential answers you identify here, the whole infinity of them, are all as good as each other (but somehow not as bad as each other).
But you say these are all valid answers, and this even though they are contradictory to each other. Me I thought a contradictory conclusion was a clue that something was wrong in the premises. Or perhaps the paradox here is meant to challenge our view of logic?

For me, though, since any two of your potential answers are contradictory to each other, only one answer can possibly do and I definitely don’t accept

\(\sum_{n=1}^\infty(1 - 1)\)

as a valid representation of a process where each steps puts ten balls in the vase and removes one. The only one proper representation I see is

\(\sum_{n=1}^\infty 9\)

which obviously is infinite and because of that there is no answer to the question of how many balls are in the vase at 12:00am. So it’s not that there is somehow infinitely many answers all as good as each other. It’s that we can’t give any proper answer. All that we can say is that there is no upper limit to the number of balls in the vase in the continuous time interval from11:00am included to 12:00am excluded.
Perhaps the conclusion is that the kind of world necessary to frame your paradox isn’t possible at all, which wouldn’t be such a big surprise.
So, according to you, how many balls are left in the vase at 12:00am again?
You’re right, you’ve said what you had to say. I guess we’re not talking about the same thing at all. You think it’s a math problem. I see it as a problem of representation. Not just the balls-&-vase thingy but for example the infinity implicit in the alleged continuity of time and space.
EB

 

[Malintent]

This is not a paradox. it is an exercise in asymptotic infinities.

The answer is that you will never reach 12:00, as the described function is asymptotic to that limit.