Paradox!

[Tom Sawyer]

The party is over you missed it....

Well, you never really miss the party. If the time differential of each subsequent step towards the end of the party is cut in half, you never actually arrive at the end of the party unless you decide to cheat and randomly say that you've taken an infinite number of steps towards that end for no good reason.

 

[Kharakov]

So is there a party line that you are suppose to cross? Because as far as I know, parties aren't clearly defined, especially after 3am.

 

[steve_bnk]

The party is over you missed it....

Well, you never really miss the party. If the time differential of each subsequent step towards the end of the party is cut in half, you never actually arrive at the end of the party unless you decide to cheat and randomly say that you've taken an infinite number of steps towards that end for no good reason.

You must like cold stale pizza.

 

[Tom Sawyer]

No. I actually have a microwave, so I can just zap it whenever I like.

 

[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!

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?

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.

Also, the process itself is not a paradox contrary to what beero1000 seems to say in the OP. To see a paradox, you need to look at his second post:

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.

Assuming that most people would claim that more balls are added at each step of the process and that therefore we can only end up with at least a very large number, possibly an infinite number, of balls, the paradox is beero1000’s reasoning purporting to show that instead of some very large number of balls we may end up with zero balls in the vase. This is the last sentence of his second post but he was already talking of a paradox in the first!

Once you understand that the process is not defined for 12:00am the paradox disappears though. At each new step, starting from zero à 11:00am, there are 9 balls more, ad infinitum. Each step gets closer in time to 12:00am without ever getting there. The total number of balls left in the vase never gets back to zero. While at 12:00am, the number of balls in the vase is not specified so the question does not make any sense at all.

The only sense it may seem to make is due to the ambiguity entertained by the wording of the paradox between a process occurring in the real world and some unrealistic mathematical model. But once this is clear, this impression also vanishes in a puff of smoke.
EB

 

[beero1000]

And we're off...

No. I actually have a microwave, so I can just zap it whenever I like.

Microwaved pizza? Ew.

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.

Also, the process itself is not a paradox contrary to what beero1000 seems to say in the OP. To see a paradox, you need to look at his second post:

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.

Assuming that most people would claim that more balls are added at each step of the process and that therefore we can only end up with at least a very large number, possibly an infinite number, of balls, the paradox is beero1000’s reasoning purporting to show that instead of some very large number of balls we may end up with zero balls in the vase. This is the last sentence of his second post but he was already talking of a paradox in the first!

Huh? I did say that there were 42 balls left at noon in my first post. In fact, the same reasoning can be used to argue that any number remains.

Once you understand that the process is not defined for 12:00am the paradox disappears though. At each new step, starting from zero à 11:00am, there are 9 balls more, ad infinitum. Each step gets closer in time to 12:00am without ever getting there. The total number of balls left in the vase never gets back to zero. While at 12:00am, the number of balls in the vase is not specified so the question does not make any sense at all.

The only sense it may seem to make is due to the ambiguity entertained by the wording of the paradox between a process occurring in the real world and some unrealistic mathematical model. But once this is clear, this impression also vanishes in a puff of smoke.
EB

I guess I shouldn't have posted in the Real World forum. If only there was a forum for Logic, then I'd be set...

 

[Tom Sawyer]

I guess I shouldn't have posted in the Real World forum. If only there was a forum for Logic, then I'd be set...

But the problem occurs when people try and relate the logical problem to the real world. They create a theoretical model which is disconnected from reality where one can have a countable infinity and then step outside of the model and try to relate the conclusions of it back to reality. When the conclusions from that theoretical model are at odds with what one would find in reality, it's then called a paradox as opposed to a limitation of the model which throws an error when trying to relate it to reality.

 

[Kharakov]

Also, the process itself is not a paradox contrary to what beero1000 seems to say in the OP. To see a paradox, you need to look at his second post:

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.

Assuming that most people would claim that more balls are added at each step of the process and that therefore we can only end up with at least a very large number, possibly an infinite number, of balls, the paradox is beero1000’s reasoning purporting to show that instead of some very large number of balls we may end up with zero balls in the vase. This is the last sentence of his second post but he was already talking of a paradox in the first!

Huh? I did say that there were 42 balls left at noon in my first post. In fact, the same reasoning can be used to argue that any number remains.

I do like the no balls at noon idea. Although only a person with ambiguously defined iterations would have no balls at noon.

In other words, one could have specific infinites, in which iterations equals a specific  hyperreal number H, in which case you have 9H balls at noon.

Light speed cutoff.

Assuming .1 meters to travel into and out of the jar, at some point one would have to exceed the speed of light. So at the point where one has to do the deed in under 1/2,997,924,580 of a second you've reached the light speed cut off.
1800 seconds / 2^42 iterations = 4.0927261579781771 *10^-10 seconds
.1 meters / 299,792,458 meters per second = 3.3356409519815207 * 10^-10 seconds

Any more iterations would take faster than light speed. So we do end up with 2^42 * 9 balls, which is where the 42 originates.

 

[Deepak]

Also, the process itself is not a paradox contrary to what beero1000 seems to say in the OP. To see a paradox, you need to look at his second post:

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.

Assuming that most people would claim that more balls are added at each step of the process and that therefore we can only end up with at least a very large number, possibly an infinite number, of balls, the paradox is beero1000’s reasoning purporting to show that instead of some very large number of balls we may end up with zero balls in the vase. This is the last sentence of his second post but he was already talking of a paradox in the first!

Huh? I did say that there were 42 balls left at noon in my first post. In fact, the same reasoning can be used to argue that any number remains.

I do like the no balls at noon idea. Although only a person with ambiguously defined iterations would have no balls at noon.

In other words, one could have specific infinites, in which iterations equals a specific  hyperreal number H, in which case you have 9H balls at noon.

I'm not understanding why we'd need the hyperreal numbers here.

I can construct a numbering system and a selection system so that I get any desired number of remaining balls simply using the non-negative integers.

 

[Kharakov]

I'm not understanding why we'd need the hyperreal numbers here.

I can construct a numbering system and a selection system so that I get any desired number of remaining balls simply using the non-negative integers.

Specifying the number of iterations as H allows one to arrive at the conclusion that 10H - H= 9H balls are in the vase at noon.

 

[steve_bnk]

And we're off...



Microwaved pizza? Ew.

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.

Also, the process itself is not a paradox contrary to what beero1000 seems to say in the OP. To see a paradox, you need to look at his second post:

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.

Assuming that most people would claim that more balls are added at each step of the process and that therefore we can only end up with at least a very large number, possibly an infinite number, of balls, the paradox is beero1000’s reasoning purporting to show that instead of some very large number of balls we may end up with zero balls in the vase. This is the last sentence of his second post but he was already talking of a paradox in the first!

Huh? I did say that there were 42 balls left at noon in my first post. In fact, the same reasoning can be used to argue that any number remains.

Once you understand that the process is not defined for 12:00am the paradox disappears though. At each new step, starting from zero à 11:00am, there are 9 balls more, ad infinitum. Each step gets closer in time to 12:00am without ever getting there. The total number of balls left in the vase never gets back to zero. While at 12:00am, the number of balls in the vase is not specified so the question does not make any sense at all.

The only sense it may seem to make is due to the ambiguity entertained by the wording of the paradox between a process occurring in the real world and some unrealistic mathematical model. But once this is clear, this impression also vanishes in a puff of smoke.
EB

I guess I shouldn't have posted in the Real World forum. If only there was a forum for Logic, then I'd be set...

How did you derive your exact number of steps?

 

[beero1000]

One more try because I really can't help myself.

To have k balls remaining at noon: For the first k removals, remove balls labeled 1,2,...,k. For subsequent removals, remove balls 2k+1, 2k+2, ..., etc. Balls k+1, ..., 2k are not removed at any time before noon, and so must be in the vase at noon. Every other ball placed before noon is removed before noon (a specific removal time for each ball is easy to calculate), and therefore cannot be in the vase at noon. No balls are placed in the vase at noon. Therefore, there are exactly k balls in the vase at noon. This is equally valid for every choice of k.

Before anyone decides to jump in with the same tired responses,

1. I know that it shouldn't be simultaneously true for every choice of k. The fact that we have valid arguments yielding absurd answers is why it's called a paradox.
2. This was never a 'real world' question.
3. Noon definitely arrives, exactly 1 hour after 11:00. Zeno's paradoxes have been resolved for a very long time.
4. Just because a function tends to infinity at a point in the limit does not mean the value at the point is infinite.
5. Just because you can argue that the number of balls at noon must be something different does not resolve the paradox.
6. There are many other choices of labeling, but the existence of the ones above is already a problem.

 

[Tom Sawyer]

The fact that we have valid arguments yielding absurd answers doesn't make it a paradox. It means that it exposes a flaw in the model. Given that this flaw occurs when you try to relate it to something real by stepping outside of the model, you can't simultaneously call it a paradox and also write it off as not being a 'real world' question.

 

[beero1000]

The fact that we have valid arguments yielding absurd answers doesn't make it a paradox. It means that it exposes a flaw in the model. Given that this flaw occurs when you try to relate it to something real by stepping outside of the model, you can't simultaneously call it a paradox and also write it off as not being a 'real world' question.

That's the definition of a paradox.

par·a·dox
ˈparəˌdäks/
noun

• a statement or proposition that, despite sound (or apparently sound) reasoning from acceptable premises, leads to a conclusion that seems senseless, logically unacceptable, or self-contradictory.

All paradoxes have resolutions because people have worked hard to make sure our logic is consistent, but claiming that it doesn't make sense and is therefore ridiculous and flawed is not actually resolving the problem. There is an inconsistency in the problem, there has to be in order for mutually contradictory results to arise in a (presumably) consistent system. You seem to be fixating on the 'relating to something real' idea, but that has nothing to do with the inconsistency. The statement would work just as well stated in abstract set theoretical terms. The problem is peoples' intuitions. You think you know what should happen, and an argument for something else seems so utterly ridiculous that you're rejecting it out of hand, even though it's just as logically valid.

 

[Tom Sawyer]

But it never leads to any logically unacceptable or self-contradictory conclusions. That's why it's not a paradox. The model just keeps adding and removing items without any end. It's only when you leave the model and look at where it would end, if it ever did, that you find a different conclusion. That's fine because the model doesn't have an end and you have to ignore the model in order to get an end to it. The model just plain doesn't tell you what you get at the end of it because that's not something it ever reaches.

 

[beero1000]

But it never leads to any logically unacceptable or self-contradictory conclusions. That's why it's not a paradox. The model just keeps adding and removing items without any end. It's only when you leave the model and look at where it would end, if it ever did, that you find a different conclusion. That's fine because the model doesn't have an end and you have to ignore the model in order to get an end to it. The model just plain doesn't tell you what you get at the end of it because that's not something it ever reaches.

That opinion went out of date around the 1890s. There are numbers beyond all of the integers, and you can fit an infinite number of steps in a finite amount of time. Zeno really is dead, I promise.

 

[Speakpigeon]

That's the definition of a paradox.

par·a·dox
ˈparəˌdäks/
noun

• a statement or proposition that, despite sound (or apparently sound) reasoning from acceptable premises, leads to a conclusion that seems senseless, logically unacceptable, or self-contradictory.

You definition is missing one relevant sense (sense 4):

Paradox
3. An assertion that is essentially self-contradictory, though based on a valid deduction from acceptable premises.
4. A statement contrary to received opinion.
Latin paradoxum, from Greek paradoxon, from neuter sing. of paradoxos, conflicting with expectation : para-, beyond; see para-1 + doxa, opinion (from dokein, to think; see dek- in Indo-European roots).

From the Latin, we also understand that definition 3 is a derivation from sense 4. And we are normally all of the opinion that the conclusion of valid deduction from acceptable premises should not be self-contradictory so one that is is a paradox in sense 4 of the word. Then some people extended to use of "paradox" with sense 3.
That's why...
EB

 

[Speakpigeon]

One more try because I really can't help myself.
<snip>
This was never a 'real world' question.

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

 

[Tom Sawyer]

But it never leads to any logically unacceptable or self-contradictory conclusions. That's why it's not a paradox. The model just keeps adding and removing items without any end. It's only when you leave the model and look at where it would end, if it ever did, that you find a different conclusion. That's fine because the model doesn't have an end and you have to ignore the model in order to get an end to it. The model just plain doesn't tell you what you get at the end of it because that's not something it ever reaches.

That opinion went out of date around the 1890s. There are numbers beyond all of the integers, and you can fit an infinite number of steps in a finite amount of time. Zeno really is dead, I promise.

Appeals to authority as as much of a logical fallacy as the rest of your thought processes here. You can't count to infinity, so the number of steps never ends.

 

[beero1000]

One more try because I really can't help myself.
<snip>
This was never a 'real world' question.

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.

That opinion went out of date around the 1890s. There are numbers beyond all of the integers, and you can fit an infinite number of steps in a finite amount of time. Zeno really is dead, I promise.

Appeals to authority as as much of a logical fallacy as the rest of your thought processes here. You can't count to infinity, so the number of steps never ends.

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?