Paradox!

[Tom Sawyer]

I matched your assertions with my own, except mine are based on logical deductions and yours are based on your intuition and amateurish understanding of Zeno's paradoxes. It is only a perk, and not the basis of my argument, to add that virtually every mathematician and philosopher of the last 100+ years has supported my argument and not yours. In fact, you don't seem to be willing to make any argument at all except 'you can't do that', albeit more and more stringently.

Is motion theoretically possible in a continuous universe? There are infinitely many incremental motions needed to move any distance. Is it possible to theoretically trace out a curve? There are infinitely many points that must be reached to do so. By your reasoning, none of these is possible. Is that really the final position you want to take?

You say that like there's some sort of consensus amongst modern matematicians which support your point of view. Others, like Pat Corvini, make the same argument that I'm making and think that you're just as wrong as I'm saying you are.

Zeno's paradox makes a subtle yet fatal switch between the physical and the abstract. The basis of the paradox is the two premises that you must traverse an infinite number of steps to reach noon and that you can never make that many steps, so Zeno said that you never reach noon. The first premise, however, is a mathematical abstraction that cannot be directly applied to the second which is a statement regarding the physical world. The physical world requires a resolution amount between the steps while mathematics can use any resolution.

Zeno's paradox is a category error where the mathematical model doesn't properly relate to anything outside of itself. It doesn't reveal any kind of paradox, it just shows the limitation of the model.

 

[Kharakov]

Why don't we take the  Grandi's series approach?

Or this series.  1 − 2 + 3 − 4 + ⋯

 

[beero1000]

I matched your assertions with my own, except mine are based on logical deductions and yours are based on your intuition and amateurish understanding of Zeno's paradoxes. It is only a perk, and not the basis of my argument, to add that virtually every mathematician and philosopher of the last 100+ years has supported my argument and not yours. In fact, you don't seem to be willing to make any argument at all except 'you can't do that', albeit more and more stringently.

Is motion theoretically possible in a continuous universe? There are infinitely many incremental motions needed to move any distance. Is it possible to theoretically trace out a curve? There are infinitely many points that must be reached to do so. By your reasoning, none of these is possible. Is that really the final position you want to take?

You say that like there's some sort of consensus amongst modern matematicians which support your point of view. Others, like Pat Corvini, make the same argument that I'm making and think that you're just as wrong as I'm saying you are.

Zeno's paradox makes a subtle yet fatal switch between the physical and the abstract. The basis of the paradox is the two premises that you must traverse an infinite number of steps to reach noon and that you can never make that many steps, so Zeno said that you never reach noon. The first premise, however, is a mathematical abstraction that cannot be directly applied to the second which is a statement regarding the physical world. The physical world requires a resolution amount between the steps while mathematics can use any resolution.

Zeno's paradox is a category error where the mathematical model doesn't properly relate to anything outside of itself. It doesn't reveal any kind of paradox, it just shows the limitation of the model.

No way! You found one person who agrees with you? And that person has a Ph.D. in ... Electrical Engineering? Are you sure you aren't in a climate change denial debate?

Sure you could even find a couple of finitist mathematicians, even ultrafinitist ones (I can point you to them if you'd like). In no way does that mean that there is not an overwhelming consensus among mathematicians. I assure you, as a mathematician, there is such a consensus among mathematicians, and (according to wiki) even among philosophers.

I noticed you fell back to your 'real world' hobby horse, even after I explicitly said that nothing about the scenario is intended to be anything other than abstract. I also noticed that you didn't respond to any of the questions I asked.

 

[Kharakov]

Quick question:

In Zeno's paradox, is the velocity of approach halved with each step, by decreasing step size and stepping at the same rate?
Or is the velocity of approach the same, and the step size decreased, so that the person starts stepping really fast?

At some point in time, wouldn't quantum uncertainty in position basically guarantee that a person crossed the finish like (once the step size is well below uncertainty in position)?

 

[Speakpigeon]

I agree it’s too late now for addressing the details of this shameless orgy of poorly worded arguments but I think it’s a good idea to provide a graceful conclusion so that innocent people reading the thread don’t feel too puzzled!



It’s clear that the process as specified above never gets to 12:00am so the question of how many balls remain in the vase at this time is just nonsensical.

I guess you think movement is impossible too.

I didn't have to use this hypothesis...

Well, it's plenty interesting. Can you explain what is movement? Can anybody? If not, how movement can be relevant to your paradox?

Now, the only notion of movement with some legitimacy, according to me, is that which we acquire with our experience of the material world, everything else is speculative. Since you have confirmed that this "paradox" is entirely theoretical, the kind of real-world movement we may have reason to believe exists is relegated to mere example of what movement could be taken to be in the theoretical world of your paradox. So I'm not denying that movement exists in the material world but you can't argue from this foreign notion without explaining why it would be relevant in the world of your paradox. Me, I think it's irrelevant but maybe you have an argument?
EB

 

[Speakpigeon]

So what is the ground for insisting that the process described has a step at 12:00am? How do we get from the steps which are specified in the OP all before 12:00am to this hypothetical last step that would occur at 12:00am?

I say that according to your specifications there is no step at 12:00am.

So the number of balls left in the vase is not defined at 12:00am.

So it's not that we don't know how many balls or that all answers are possible. No, it's just that the question is absurd, like asking how many pinguins are in the vase even though we have no idea how they would have gotten into it.

Ok, that being said, it's an interesting paradox. However, you don't seem to be prepared to argue your position. I don't want to push you but I'd like to know what your argument is.
EB

There is no last step, there is no step at 12:00, and all steps happen before 12:00. That does not mean that the process never ends. Again, this is not controversial, and has not been since Cantor.

What is the last real number less than 1? Does that mean that the interval (0,1) never ends? Does it mean that it is not possible to determine any number bigger than 1? If a function is known on (0,1) is it absurd to ask for its value at 1? Just because there are an infinite number of points in an interval does not mean that the interval is unbounded.

We can tweak a little bit the parameters of your paradox to make the point more clearly. Instead of going with a time step with a parameter of 0.5, let's have 0.9! This also have the advantage of going back to a point argued in shambolic manner by other posters. So, instead of 11:00am, 11:30am, 11:45am etc. let's have this function:

f(0) = 11 and f = f(n-1) + 9/(10^n)

So,

f(0) = 11
f(1) = 11.9
f(2) = 11.99
f(3) = 11.999
...
f(10) = 11.9999999999
...
f(20) = 11.99999999999999999999

etc.

There is a limit obviously and it's 12 (or 12:00am if you like), so the successive values of f get ever closer to 12 but there isn’t any integer n such that f = 12. Yes?

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...

You would have to explain if you think there is something else that does the same job.
EB

 

[Speakpigeon]

So what is the ground for insisting that the process described has a step at 12:00am? How do we get from the steps which are specified in the OP all before 12:00am to this hypothetical last step that would occur at 12:00am?

I say that according to your specifications there is no step at 12:00am.

So the number of balls left in the vase is not defined at 12:00am.

So it's not that we don't know how many balls or that all answers are possible. No, it's just that the question is absurd, like asking how many pinguins are in the vase even though we have no idea how they would have gotten into it.

Ok, that being said, it's an interesting paradox. However, you don't seem to be prepared to argue your position. I don't want to push you but I'd like to know what your argument is.
EB

There is no last step, there is no step at 12:00, and all steps happen before 12:00. That does not mean that the process never ends. Again, this is not controversial, and has not been since Cantor.

What is the last real number less than 1? Does that mean that the interval (0,1) never ends? Does it mean that it is not possible to determine any number bigger than 1? If a function is known on (0,1) is it absurd to ask for its value at 1? Just because there are an infinite number of points in an interval does not mean that the interval is unbounded.

I'm not arguing that.

We could consider the bounded interval. It's Ok. In my model, there is an interval of time bounded by 11:00am and 12:00am and it's possible to get continuously closer and closer to 12:00am so that for any epsilon we can always find a time t such that 12:00am - t < epsilon. If I remember it's called "compactness" (sorry, from French "compact").

So, it's good. My point instead is that I have no reason, as per your specification of your "paradox", to assume that the number of balls leftover in the vase at 12:00am is known. To me, it's just not specified. Remember that you accepted that it's not a material world situation. We have to interpret your specification as best we can. What is a "ball" in your non-realistic paradox? I have no reason to assume it's really very like an actual ball. Same for the "vase". So, as far as I can make out, "ball" is just an axis of Integers; and "the number of balls leftover after the completion of step n of the process" is a function of n, say B, taking values on this axis.

Now, as per your specification, there is no "ball" value for 12:00am because there is no integer n such that step n takes place at 12:00am sharp.

Also, the limit is not part of the set of time values of the function. Both the values and the limit belong to the same time set, and in this case a bounded time interval, say 11:00am to 12:00am, but the time values of the function are all within the unbounded interval while obviously 12:00am is at the boundary. The two sets are therefore disconnected, so no "ball" value for 12:00am, as per your specification.

There is some more to say but let's see if you can explain your paradoxical point of view.
EB

 

[Juma]

There is a limit obviously and it's 12 (or 12:00am if you like), so the successive values of f get ever closer to 12 but there isn’t any integer n such that f = 12. Yes?

So what? Why is this a problem? If each step takes the time stipulated then when we reach 12:00 n goes beyond any number. So what? Is this somehow logically impossible? After 12:00 n is not undefined since we sre beyond the domain where n is defined.

 

[Tom Sawyer]

I noticed you fell back to your 'real world' hobby horse, even after I explicitly said that nothing about the scenario is intended to be anything other than abstract. I also noticed that you didn't respond to any of the questions I asked.

And this is the heart of your problem. There isn't any kind of paradox when the scenario is kept abstract. It's just a divergent series. The only time a "paradox" occurs is when you want to apply it to something outside of the scenario and that doesn't indicate a paradox, but rather a limitation of the scenario.

 

[beero1000]

There is no last step, there is no step at 12:00, and all steps happen before 12:00. That does not mean that the process never ends. Again, this is not controversial, and has not been since Cantor.

What is the last real number less than 1? Does that mean that the interval (0,1) never ends? Does it mean that it is not possible to determine any number bigger than 1? If a function is known on (0,1) is it absurd to ask for its value at 1? Just because there are an infinite number of points in an interval does not mean that the interval is unbounded.

We can tweak a little bit the parameters of your paradox to make the point more clearly. Instead of going with a time step with a parameter of 0.5, let's have 0.9! This also have the advantage of going back to a point argued in shambolic manner by other posters. So, instead of 11:00am, 11:30am, 11:45am etc. let's have this function:

f(0) = 11 and f = f(n-1) + 9/(10^n)

So,

f(0) = 11
f(1) = 11.9
f(2) = 11.99
f(3) = 11.999
...
f(10) = 11.9999999999
...
f(20) = 11.99999999999999999999

etc.

There is a limit obviously and it's 12 (or 12:00am if you like), so the successive values of f get ever closer to 12 but there isn’t any integer n such that f = 12. Yes?

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...

You would have to explain if you think there is something else that does the same job.
EB

This is actually really close to the actual resolution of the paradox. The idea is that people's intuition is that the function describing the number of balls in the vase is continuous at infinity. This is true for any finite number, but not true for infinity. For any function continuous at a, \(\lim_{x\to a} f(x) = f(a)\). The question asks (essentially) for \(f(\infty}\), and people assume that that is equal to \(\lim_{x \to \infty} f(x)\). Since, this is not true we have a problem. In fact, the question does not give enough information to determine \(f(\infty)\), which is why there are multiple possible (and simultaneously correct) values.

There is no last step, there is no step at 12:00, and all steps happen before 12:00. That does not mean that the process never ends. Again, this is not controversial, and has not been since Cantor.

What is the last real number less than 1? Does that mean that the interval (0,1) never ends? Does it mean that it is not possible to determine any number bigger than 1? If a function is known on (0,1) is it absurd to ask for its value at 1? Just because there are an infinite number of points in an interval does not mean that the interval is unbounded.

I'm not arguing that.

We could consider the bounded interval. It's Ok. In my model, there is an interval of time bounded by 11:00am and 12:00am and it's possible to get continuously closer and closer to 12:00am so that for any epsilon we can always find a time t such that 12:00am - t < epsilon. If I remember it's called "compactness" (sorry, from French "compact").

So, it's good. My point instead is that I have no reason, as per your specification of your "paradox", to assume that the number of balls leftover in the vase at 12:00am is known. To me, it's just not specified. Remember that you accepted that it's not a material world situation. We have to interpret your specification as best we can. What is a "ball" in your non-realistic paradox? I have no reason to assume it's really very like an actual ball. Same for the "vase". So, as far as I can make out, "ball" is just an axis of Integers; and "the number of balls leftover after the completion of step n of the process" is a function of n, say B, taking values on this axis.

Now, as per your specification, there is no "ball" value for 12:00am because there is no integer n such that step n takes place at 12:00am sharp.

Also, the limit is not part of the set of time values of the function. Both the values and the limit belong to the same time set, and in this case a bounded time interval, say 11:00am to 12:00am, but the time values of the function are all within the unbounded interval while obviously 12:00am is at the boundary. The two sets are therefore disconnected, so no "ball" value for 12:00am, as per your specification.

There is some more to say but let's see if you can explain your paradoxical point of view.
EB

Not compactness, the property that you describe means 12:00 is a limit point of the the set of times.

You are essentially there, except possibly for semantics. Instead of saying that there is no ball value, I would say that any ball value is possible, so the question of 'how many balls are there' is not yet well-defined. The reason I think that this is a better distinction is that if a labeling scheme is specified the inconsistency goes away and there is only one possible answer (even though the semantic problems of 'ball' and 'vase' remain).

I noticed you fell back to your 'real world' hobby horse, even after I explicitly said that nothing about the scenario is intended to be anything other than abstract. I also noticed that you didn't respond to any of the questions I asked.

And this is the heart of your problem. There isn't any kind of paradox when the scenario is kept abstract. It's just a divergent series. The only time a "paradox" occurs is when you want to apply it to something outside of the scenario and that doesn't indicate a paradox, but rather a limitation of the scenario.

Yes there is. Unless you think 0 = 1?

 

[steve_bnk]

Beero,


Post 1

'...Start with an empty vase at 11:00am. At 11:30am, place 10 balls in the vase, and remove one ball. At 11:45am, place 10 more balls in the vase and remove one ball. Continue repeating the procedure of adding 10 balls and removing one, but at each step reduce the time between steps by half. Question: At noon, how many balls are in the vase?

My answer: At noon, there are exactly 42 balls in the vase.

Of course, there are many other possible answers; this is a paradox, after all. Thoughts?..'



Post 4

'….I never said that you have to remove one of the ten just placed. Adding ten and removing one isn't the same thing as adding nine.

For example, if we imagine placing balls 1 - 10, and then removing ball 1 at 11:30am, then placing balls 11-20 and removing ball 2 at 11:45am, etc. It should be clear that every ball placed in the vase will have some number label, and also that every number label is removed at some point before noon. Therefore, all balls have been removed by noon, and so the vase is empty. '

Arithmetic depends on when balls are added or how the balls are numbered?

'...Start with an empty vase at 11:00am, at 11:30am add a (countable) infinity of balls to the vase and remove 1 ball...'

'...Sure, that's the paradox. The number of balls in the vase keeps increasing, but so does the number of balls removed. The intuition says 'but balls are being placed faster than they are being removed', but why does that matter as long as every placed ball is removed before noon?..'

Are you serious?

'...As for how to get 42, place balls 1 - 10 at the first step and remove ball 1, place balls 11-20 at the second step and remove ball 2, place balls 21-30 and remove ball 3, place balls 31-40 and remove ball 4, place balls 41-50 and remove ball 5, then place balls 51-60 and remove ball 47, place 61-70 and remove ball 48, etc. Every ball except 5, 6, ..., 45, 46 is removed, so those are exactly the balls remaining. Therefore, there are exactly 42 balls in the vase at noon...'


A paradox is when there are mutually exclusive conditions that ca not be resolved. You have not shown how you get to 12:00 by taking infinitely small steps.

In the last quote above how does numbering the balls override arithmetic, add 10 and take away one.

place balls 1 - 10 at the first step and remove ball 1 – total 9
place balls 11-20 at the second step and remove ball 2, - total 18
place balls 21-30 and remove ball 3 – total 27

You had a brain fart in your logic, it happens to everyone . You need to rethink it. Your logic is itself a paradox...

You still have not answered my question, show the first 10 iterations and time steps of your solution. I suspect you can not.

 

[beero1000]

Beero,


Post 1

'...Start with an empty vase at 11:00am. At 11:30am, place 10 balls in the vase, and remove one ball. At 11:45am, place 10 more balls in the vase and remove one ball. Continue repeating the procedure of adding 10 balls and removing one, but at each step reduce the time between steps by half. Question: At noon, how many balls are in the vase?

My answer: At noon, there are exactly 42 balls in the vase.

Of course, there are many other possible answers; this is a paradox, after all. Thoughts?..'



Post 4

'….I never said that you have to remove one of the ten just placed. Adding ten and removing one isn't the same thing as adding nine.

For example, if we imagine placing balls 1 - 10, and then removing ball 1 at 11:30am, then placing balls 11-20 and removing ball 2 at 11:45am, etc. It should be clear that every ball placed in the vase will have some number label, and also that every number label is removed at some point before noon. Therefore, all balls have been removed by noon, and so the vase is empty. '

Arithmetic depends on when balls are added or how the balls are numbered?

'...Start with an empty vase at 11:00am, at 11:30am add a (countable) infinity of balls to the vase and remove 1 ball...'

'...Sure, that's the paradox. The number of balls in the vase keeps increasing, but so does the number of balls removed. The intuition says 'but balls are being placed faster than they are being removed', but why does that matter as long as every placed ball is removed before noon?..'

Are you serious?

'...As for how to get 42, place balls 1 - 10 at the first step and remove ball 1, place balls 11-20 at the second step and remove ball 2, place balls 21-30 and remove ball 3, place balls 31-40 and remove ball 4, place balls 41-50 and remove ball 5, then place balls 51-60 and remove ball 47, place 61-70 and remove ball 48, etc. Every ball except 5, 6, ..., 45, 46 is removed, so those are exactly the balls remaining. Therefore, there are exactly 42 balls in the vase at noon...'


A paradox is when there are mutually exclusive conditions that ca not be resolved. You have not shown how you get to 12:00 by taking infinitely small steps.

In the last quote above how does numbering the balls override arithmetic, add 10 and take away one.

place balls 1 - 10 at the first step and remove ball 1 – total 9
place balls 11-20 at the second step and remove ball 2, - total 18
place balls 21-30 and remove ball 3 – total 27

You had a brain fart in your logic, it happens to everyone . You need to rethink it. Your logic is itself a paradox...

You still have not answered my question, show the first 10 iterations and time steps of your solution. I suspect you can not.

Infinite limits do funny things with arithmetic. The sum of two rational numbers is rational. The sum of an infinite number of rational numbers may not be.

I have posted way too many explanations already, and they are all available in-thread. Work it out yourself. The calculations are not hard.

 

[Juma]

Instead of saying that there is no ball value, I would say that any ball value is possible, so the question of 'how many balls are there' is not yet well-defined. The reason I think that this is a better distinction is that if a labeling scheme is specified the inconsistency goes away and there is only one possible answer (even though the semantic problems of 'ball' and 'vase' remain).

<hunch>
The center of the paradox is not the labelling scheme per se. It is the failure to se this as the multidimensional problem it really is.
By letting one parameter go to infinity in a limited set you actually introduce a new (fractional) dimension. At 12 the dimension with balls has gone off in an ortogonal direction.

The center of the paradox is that the question "how many balls are there" suggest a single time and place but the actual problem requires two dimensions of time.</hunch>

 

[steve_bnk]

Beero,


Post 1

'...Start with an empty vase at 11:00am. At 11:30am, place 10 balls in the vase, and remove one ball. At 11:45am, place 10 more balls in the vase and remove one ball. Continue repeating the procedure of adding 10 balls and removing one, but at each step reduce the time between steps by half. Question: At noon, how many balls are in the vase?

My answer: At noon, there are exactly 42 balls in the vase.

Of course, there are many other possible answers; this is a paradox, after all. Thoughts?..'



Post 4

'….I never said that you have to remove one of the ten just placed. Adding ten and removing one isn't the same thing as adding nine.

For example, if we imagine placing balls 1 - 10, and then removing ball 1 at 11:30am, then placing balls 11-20 and removing ball 2 at 11:45am, etc. It should be clear that every ball placed in the vase will have some number label, and also that every number label is removed at some point before noon. Therefore, all balls have been removed by noon, and so the vase is empty. '

Arithmetic depends on when balls are added or how the balls are numbered?

'...Start with an empty vase at 11:00am, at 11:30am add a (countable) infinity of balls to the vase and remove 1 ball...'

'...Sure, that's the paradox. The number of balls in the vase keeps increasing, but so does the number of balls removed. The intuition says 'but balls are being placed faster than they are being removed', but why does that matter as long as every placed ball is removed before noon?..'

Are you serious?

'...As for how to get 42, place balls 1 - 10 at the first step and remove ball 1, place balls 11-20 at the second step and remove ball 2, place balls 21-30 and remove ball 3, place balls 31-40 and remove ball 4, place balls 41-50 and remove ball 5, then place balls 51-60 and remove ball 47, place 61-70 and remove ball 48, etc. Every ball except 5, 6, ..., 45, 46 is removed, so those are exactly the balls remaining. Therefore, there are exactly 42 balls in the vase at noon...'


A paradox is when there are mutually exclusive conditions that ca not be resolved. You have not shown how you get to 12:00 by taking infinitely small steps.

In the last quote above how does numbering the balls override arithmetic, add 10 and take away one.

place balls 1 - 10 at the first step and remove ball 1 – total 9
place balls 11-20 at the second step and remove ball 2, - total 18
place balls 21-30 and remove ball 3 – total 27

You had a brain fart in your logic, it happens to everyone . You need to rethink it. Your logic is itself a paradox...

You still have not answered my question, show the first 10 iterations and time steps of your solution. I suspect you can not.

Infinite limits do funny things with arithmetic. The sum of two rational numbers is rational. The sum of an infinite number of rational numbers may not be.

I have posted way too many explanations already, and they are all available in-thread. Work it out yourself. The calculations are not hard.

True to form.. when challenged you resort to dismissiveness and hand waving, anything but directly answering a challenge to your assertions.

 

[beero1000]

Infinite limits do funny things with arithmetic. The sum of two rational numbers is rational. The sum of an infinite number of rational numbers may not be.

I have posted way too many explanations already, and they are all available in-thread. Work it out yourself. The calculations are not hard.

True to form.. when challenged you resort to dismissiveness and hand waving, anything but directly answering a challenge to your assertions.

Ridiculous challenges that are, as usual, completely irrelevant to the problem at hand, yes. I am well aware that your math skills are subpar, but I've already specified the labeling\removal procedures and 12 - 1/2^N is not a difficult sequence to calculate. The question as to why you'd think listing the first 10 iterations would possibly be helpful or informative still remains.

 

[Speakpigeon]

There is a limit obviously and it's 12 (or 12:00am if you like), so the successive values of f get ever closer to 12 but there isn’t any integer n such that f = 12. Yes?

So what? Why is this a problem? If each step takes the time stipulated then when we reach 12:00 n goes beyond any number. So what? Is this somehow logically impossible? After 12:00 n is not undefined since we sre beyond the domain where n is defined.

No one cares whether "we" reach 12:00am or not.

Remember, this is not a physical process, as per beero1000's own admission. So, we have no reason whatsoever to assume that what may apply in the so called "real world" applies ipso facto in the theoretical world specified by him.

So what matters is that the process defined in the OP should reach 12:00am if we are to know how many balls are left over at 12:00am. We would need one step of the process to occur at 12:00am and none does. Full stop.
EB

 

[Speakpigeon]

Instead of saying that there is no ball value, I would say that any ball value is possible, so the question of 'how many balls are there' is not yet well-defined. The reason I think that this is a better distinction is that if a labeling scheme is specified the inconsistency goes away and there is only one possible answer (even though the semantic problems of 'ball' and 'vase' remain).

<hunch>
The center of the paradox is not the labelling scheme per se. It is the failure to se this as the multidimensional problem it really is.
By letting one parameter go to infinity in a limited set you actually introduce a new (fractional) dimension. At 12 the dimension with balls has gone off in an ortogonal direction.

The center of the paradox is that the question "how many balls are there" suggest a single time and place but the actual problem requires two dimensions of time.</hunch>

Yes, it's spooky but I just had EXACTLY the same psychedelic experience right in the middle of Wednesday night.

Well, I think I had way too much cacao and sardine just before going to bed. A bit heavy on the stomach I guess!

I don't want you to incriminate yourself but what do you use?
EB

 

[Speakpigeon]

This is actually really close to the actual resolution of the paradox. The idea is that people's intuition is that the function describing the number of balls in the vase is continuous at infinity.

At infinity?!

You are aware that infinity just means "without bound", "not bounded". So, saying "at infinity" is a bit like saying "inside nothingness".

How people's intuition could possibly encompass the concept of "at infinity"?

Rather, you specified a process, clearly enough, which just keeps adding 9 balls at every step before 12:00am. So it's not like it's difficult to do the maths! And that's just what people do.

Let's simplify the problem though. adding 9 balls may well be a tad too hard. Let's add 1 ball at a time. So let's have the function:

f(0)=0 and f = f(n - 1) + 1 for n > 0

Easy enough, yes?

So we have:

f(0) = 0
f(1) = 1
f(2) = 2
f(3) = 3
...
f(10) = 10
...
f(20) = 20
etc.

So, basically, without being a maths genius, we can see that the resulting set is N itself, yes? So f = n.

Let's also assume that each f step occurs at the time specified in the OP. So, now, what is the value of f at 12:00am?

Me, I say the same thing as before, we just don't know because the question is meaningless because there's no step occurring at 12:00am as per your specification.

Now, if you reply "infinity", let's hear your argument (or demonstration).

In fact, the question does not give enough information to determine \(f(\infty)\), which is why there are multiple possible (and simultaneously correct) values.

Sure, when we don't know x then we accept that x may take any particular value if there is a reason that it should take a particular value. But absent any reason for x to have a particular value, the question of what value it has is just meaningless.

In the material world, we assume unknown things to have particular characteristics but you accepted that we're not talking about the material world so there is no reason that the number of balls left over after the completion of each step of the process you specified should take a particular value and therefore the question of what value it has is just meaningless.
EB

 

[Speakpigeon]

The idea is that people's intuition is that the function describing the number of balls in the vase is continuous at infinity. This is true for any finite number, but not true for infinity. For any function continuous at a, \(\lim_{x\to a} f(x) = f(a)\).

Oh come on, you didn't specify f(a), that is, f(12:00am)! Nobody is assuming anything about f(12:00am) because you didn't specify f(12:00am).

The question asks (essentially) for \(f(\infty}\), and people assume that that is equal to \(\lim_{x \to \infty} f(x)\). Since, this is not true we have a problem.

The problem would be the same without infinity to make the maths hard to intuite.

In fact, the question does not give enough information to determine \(f(\infty)\).

The OP does not give enough information to determine f(12:00am). Infinity is irrelevant here. That's it. There's nothing else to say about it! It's not that f(12:00am) has a value we don't know or that we have vox populi delusions about it. It's that the question of the value of f(12:00am) is meaningless to start with.
EB

 

[beero1000]

At infinity?!

You are aware that infinity just means "without bound", "not bounded". So, saying "at infinity" is a bit like saying "inside nothingness".

Annnndddd... we'e back to pre-calculus. You are aware that there are many different kinds of infinity? Some larger than others? That there are standard number systems with infinity as a point?

How people's intuition could possibly encompass the concept of "at infinity"?

Apparently, with great difficulty.

Rather, you specified a process, clearly enough, which just keeps adding 9 balls at every step before 12:00am. So it's not like it's difficult to do the maths! And that's just what people do.

Let's simplify the problem though. adding 9 balls may well be a tad too hard. Let's add 1 ball at a time. So let's have the function:

f(0)=0 and f = f(n - 1) + 1 for n > 0

Easy enough, yes?

So we have:

f(0) = 0
f(1) = 1
f(2) = 2
f(3) = 3
...
f(10) = 10
...
f(20) = 20
etc.

So, basically, without being a maths genius, we can see that the resulting set is N itself, yes? So f = n.

Wrong. You're regressing here. If I tell you that f(x) = x for all x < 1, you cannot say that f(1) = 1.

Let's also assume that each f step occurs at the time specified in the OP. So, now, what is the value of f at 12:00am?

Me, I say the same thing as before, we just don't know because the question is meaningless because there's no step occurring at 12:00am as per your specification.

Nonsense, there's no reason to require that there be a step at 12.

Now, if you reply "infinity", let's hear your argument (or demonstration).

In fact, the question does not give enough information to determine \(f(\infty)\), which is why there are multiple possible (and simultaneously correct) values.

Sure, when we don't know x then we accept that x may take any particular value if there is a reason that it should take a particular value. But absent any reason for x to have a particular value, the question of what value it has is just meaningless.

In the material world, we assume unknown things to have particular characteristics but you accepted that we're not talking about the material world so there is no reason that the number of balls left over after the completion of each step of the process you specified should take a particular value and therefore the question of what value it has is just meaningless.
EB

Wrong. The same question with a labeling scheme specified is completely tame, and has a well-defined answer.

Oh come on, you didn't specify f(a), that is, f(12:00am)! Nobody is assuming anything about f(12:00am) because you didn't specify f(12:00am).

The question asks (essentially) for \(f(\infty}\), and people assume that that is equal to \(\lim_{x \to \infty} f(x)\). Since, this is not true we have a problem.

The problem would be the same without infinity to make the maths hard to intuite.

In fact, the question does not give enough information to determine \(f(\infty)\).

The OP does not give enough information to determine f(12:00am). Infinity is irrelevant here. That's it. There's nothing else to say about it! It's not that f(12:00am) has a value we don't know or that we have vox populi delusions about it. It's that the question of the value of f(12:00am) is meaningless to start with.
EB

You're right that the statement does not give enough information to define f(12am). You're wrong in claiming meaninglessness of f(12am). It's completely reasonable to ask for the value of a function at a point for which it's defined. Meaningful too.