Paradox!

[Tom Sawyer]

Imagine spending an hour drawing a 100m straight line.

In the first half hour you draw a 50m segment of the line.

In the next quarter hour you draw a 25m segment of the line.

In the next 7.5 minutes you draw a 12.m segment of the line.

Etc. etc.

For each subsequently half of the previous time you draw a correspondingly segment of the line half as long as the previous segment.

How long is the line after an hour has passed?

It doesn't make any sense to me to say that an hour will never pass. Of course an hour will pass. An hour will pass exactly one hour after you started drawing the line. So how long is the line then?

It's the etc, etc that's important.

In your scenario, how many iterations of drawing the line is it which gets you to 100m and an hour?

 

[ryan]

Imagine spending an hour drawing a 100m straight line.

In the first half hour you draw a 50m segment of the line.

In the next quarter hour you draw a 25m segment of the line.

In the next 7.5 minutes you draw a 12.m segment of the line.

Etc. etc.

For each subsequently half of the previous time you draw a correspondingly segment of the line half as long as the previous segment.

How long is the line after an hour has passed?

It doesn't make any sense to me to say that an hour will never pass. Of course an hour will pass. An hour will pass exactly one hour after you started drawing the line. So how long is the line then?

What step of ball manipulation is he performing when it becomes Noon?

He puts the last ball in at noon. It takes 0 seconds to do this.

 

[ryan]

Imagine spending an hour drawing a 100m straight line.

In the first half hour you draw a 50m segment of the line.

In the next quarter hour you draw a 25m segment of the line.

In the next 7.5 minutes you draw a 12.m segment of the line.

Etc. etc.

For each subsequently half of the previous time you draw a correspondingly segment of the line half as long as the previous segment.

How long is the line after an hour has passed?

It doesn't make any sense to me to say that an hour will never pass. Of course an hour will pass. An hour will pass exactly one hour after you started drawing the line. So how long is the line then?

It's the etc, etc that's important.

In your scenario, how many iterations of drawing the line is it which gets you to 100m and an hour?

They will slowly chip away your argument with a slightly better one; your fight always loses. It happened to me, it happened to them, and it will happen to you.

Make the leap from 0.9999... to 1; it's better on this side. :devil-smiley-029:

 

[Tom Sawyer]

They will slowly chip away your argument with a slightly better one; your fight always loses. It happened to me, it happened to them, and it will happen to you.

Make the leap from 0.9999... to 1; it's better on this side. :devil-smiley-029:

You know who else thought that 0.9999 and 1 were equal? Hitler, that's who.

 

[beero1000]

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

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

Nonsense. Again, Zeno's paradoxes have been resolved for several hundred years at the very least. Noon does occur.

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

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

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

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

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

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

So what? The premises handle those too.

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

Yes, they do. Specifically, premises 4 and 5 state that since they were placed before noon, they will also be removed before noon.

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

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

Each ball is placed after a finite number of steps, because each ball is placed before noon.

Why isn't thinking about it as (1+1+1+1+1+1.....) - (1+1+1+1+1+1.....) just as valid as way as you are doing here? Isn't there a 1:1 correspondence between the balls being added and removed either way? One way of looking at it (yours) shows it as 0 balls being added infinitely many times and another way of looking at it sees it as infinitely many balls being added in an hour and infinitely many balls being removed in the same hour.

And I think this is beero's point. Our everyday fast and loose thinking about infinity leads to contradictions and paradoxes depending upon how you look at the problem.

- - - Updated - - -

That interval never actually gets down to zero.

So you're saying that an hour after 11 O'Clock it won't be noon?

That was actually my point, yet it seems to be ignored over and over again. I had actually hoped the discussion would be about resolving the paradox, not arguing about whether or not the paradox exists in the first place.

By that logic, noon never arrives in the real world because you can divide up the hour from 11 to 12 into intervals of 30 minutes, 15 minutes, 7.5 minutes, 3.75 minutes, etc. If you continue halving the difference between the current time and noon, you'll never reach noon.

But you have to halve the current time to half way to noon. And from then to the next halfway point to noon. And from then to the next halfway point. Ad infinitum.

So why does noon come and go every day if what you say is true?

Ya, that's the entire point. It's why the problem needs to be discussed as a pure math problem as opposed to a math problem which can be implemented in the real world.

This is the logic board. No one ever said anything about implementing this in the real world.

 

[beero1000]

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

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

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

5. is definitely true - given a ball with finite label n, the removal criterion means the ball is removed at exactly \(\frac{1}{2^n}\) hours before noon. The complaint that other balls have been placed simultaneously is completely irrelevant to the statement, which talks about one specific ball and whether or not it is removed before noon.

6. is also true, but could probably be worded better to avoid the case where the same ball/label is placed multiple times (just to clarify, but this already never happens due to the labeling procedure). The only complaint to this seems to be Tom Sawyer's weird ultrafinitist 'noon-never-happens' assertion.

Does anyone have an issue with these statements that isn't "my intuition doesn't agree with the conclusion these statements imply"?

 

[Tom Sawyer]

Does anyone have an issue with these statements that isn't "my intuition doesn't agree with the conclusion these statements imply"?

Just that your math is totally wrong and you're essentially arguing that 3*3=10 because you don't understand basic logic.

Other than that, not much.

 

[beero1000]

Does anyone have an issue with these statements that isn't "my intuition doesn't agree with the conclusion these statements imply"?

Just that your math is totally wrong and you're essentially arguing that 3*3=10 because you don't understand basic logic.

Other than that, not much.

I've been on this board for a while now. If you haven't seen enough of my posts by now to get that I more than understand logic and mathematics, then I'm not sure what I can do to convince you.

 

[Tom Sawyer]

Does anyone have an issue with these statements that isn't "my intuition doesn't agree with the conclusion these statements imply"?

Just that your math is totally wrong and you're essentially arguing that 3*3=10 because you don't understand basic logic.

Other than that, not much.

I've been on this board for a while now. If you haven't seen enough of my posts by now to get that I more than understand logic and mathematics, then I'm not sure what I can do to convince you.

I'm just talking about this thread. You're using a set which doesn't represent what's in that set and then calling the discrepancies resulting from the use of that set a paradox instead of a mistake. You're asserting that if you keep adding 9+9+9... you'll eventually end up at zero. You keep talking about how ball N has been removed while ignoring that every N has an N+1 and whichever N you pick will have higher numbers still there and that's always true for any value of N. You're saying that if you keep adding finite numbers together, you eventually get to infinity.

Those are fairly basic errors. You're not showing a paradox, you're just showing poor math.

 

[bigfield]

5. is definitely true - given a ball with finite label n, the removal criterion means the ball is removed at exactly \(\frac{1}{2^n}\) hours before noon. The complaint that other balls have been placed simultaneously is completely irrelevant to the statement, which talks about one specific ball and whether or not it is removed before noon.

The addition of the other balls is relevant to the problem.

I agree that every ball that is numbered is also removed before noon. But for every ball that is removed, the quantity of numbered balls remaining in the vase increases by 9.

The problem is that something has to give. In reality, it is impossible to make an infinite number of transactions in a finite length of time. In reality, 12pm must arrive and the number of transactions must cease at a finite number, much like Steve's simulation. No function dependent on time can, in practice, have an asymptote. One can't say that t=12pm must arrive because t is inexorable time and also say that an infinite number of transactions have occured in the finite length of time between 11:00am and 12:00pm. It's absurd; a contradiction.

On the Wiki page for the Ross-Littlewood Paradox:

The problem is ill-posed. To be precise, according to the problem statement, an infinite number of operations will be performed before noon, and then asks about the state of affairs at noon. But, as in Zeno's paradoxes, if infinitely many operations have to take place (sequentially) before noon, then noon is a point in time that can never be reached. On the other hand, to ask how many balls will be left at noon is to assume that noon will be reached. Hence there is a contradiction implicit in the very statement of the problem, and this contradiction is the assumption that one can somehow 'complete' an infinite number of steps. This is the solution favored by mathematician and philosopher Jean Paul Van Bendegem.

Emphasis mine.

 

[beero1000]

Does anyone have an issue with these statements that isn't "my intuition doesn't agree with the conclusion these statements imply"?

Just that your math is totally wrong and you're essentially arguing that 3*3=10 because you don't understand basic logic.

Other than that, not much.

I've been on this board for a while now. If you haven't seen enough of my posts by now to get that I more than understand logic and mathematics, then I'm not sure what I can do to convince you.

I'm just talking about this thread. You're using a set which doesn't represent what's in that set and then calling the discrepancies resulting from the use of that set a paradox instead of a mistake. You're asserting that if you keep adding 9+9+9... you'll eventually end up at zero. You keep talking about how ball N has been removed while ignoring that every N has an N+1 and whichever N you pick will have higher numbers still there and that's always true for any value of N. You're saying that if you keep adding finite numbers together, you eventually get to infinity.

Those are fairly basic errors. You're not showing a paradox, you're just showing poor math.

So I understand math, except in a thread titled "Paradox!" in which I claim to show a paradox that was constructed to show how people's intuition about infinity yields mutually contradictory answers. You think I'm showing poor math because I'm showing that there are weird answers that are completely counterintuitive, and seem impossible, when that is the entire point of the paradox - they do seem impossible, but they are just as logically consistent as the intuitive answer.

 

[ryan]

Does anyone have an issue with these statements that isn't "my intuition doesn't agree with the conclusion these statements imply"?

Just that your math is totally wrong and you're essentially arguing that 3*3=10 because you don't understand basic logic.

Other than that, not much.

I've been on this board for a while now. If you haven't seen enough of my posts by now to get that I more than understand logic and mathematics, then I'm not sure what I can do to convince you.

I'm just talking about this thread. You're using a set which doesn't represent what's in that set and then calling the discrepancies resulting from the use of that set a paradox instead of a mistake. You're asserting that if you keep adding 9+9+9... you'll eventually end up at zero. You keep talking about how ball N has been removed while ignoring that every N has an N+1 and whichever N you pick will have higher numbers still there and that's always true for any value of N. You're saying that if you keep adding finite numbers together, you eventually get to infinity.

Those are fairly basic errors. You're not showing a paradox, you're just showing poor math.

When a mathematician/beero1000 tells you this, then only a mathematical proof is going to prove you right.

You may argue with a mathematician philosophically about this, but you cannot argue the math.

 

[beero1000]

The addition of the other balls is relevant to the problem.

I agree that every ball that is numbered is also removed before noon. But for every ball that is removed, the quantity of numbered balls remaining in the vase increases by 9.

The problem is that something has to give. In reality, it is impossible to make an infinite number of transactions in a finite length of time. In reality, 12pm must arrive and the number of transactions must cease at a finite number, much like Steve's simulation. No function dependent on time can, in practice, have an asymptote. One can't say that t=12pm must arrive because t is inexorable time and also say that an infinite number of transactions have occured in the finite length of time between 11:00am and 12:00pm. It's absurd; a contradiction.

Reality has nothing to do with this, we are in the logic forum.

On the Wiki page for the Ross-Littlewood Paradox:

The problem is ill-posed. To be precise, according to the problem statement, an infinite number of operations will be performed before noon, and then asks about the state of affairs at noon. But, as in Zeno's paradoxes, if infinitely many operations have to take place (sequentially) before noon, then noon is a point in time that can never be reached. On the other hand, to ask how many balls will be left at noon is to assume that noon will be reached. Hence there is a contradiction implicit in the very statement of the problem, and this contradiction is the assumption that one can somehow 'complete' an infinite number of steps. This is the solution favored by mathematician and philosopher Jean Paul Van Bendegem.

Emphasis mine.

This is the strict finitist position. It has implications that most mathematicians reject, and most people would reject too, if they followed it to its logical conclusion. For example: Jean Paul Van Bendegem supports the notion that there is a largest integer.

 

[Tom Sawyer]

So I understand math, except in a thread titled "Paradox!" in which I claim to show a paradox that was constructed to show how people's intuition about infinity yields mutually contradictory answers. You think I'm showing poor math because I'm showing that there are weird answers that are completely counterintuitive, and seem impossible, when that is the entire point of the paradox - they do seem impossible, but they are just as logically consistent as the intuitive answer.

The reason that the answers are contradictory is because the set you're using to describe the problem isn't the same as the thing you're saying is getting put into the set. It's not a paradox, it's an error. You're taking an equation which doesn't complete and creating a set from after it's completed. You're taking an equation which is using finite operations and then calling it a paradox when infinite operations give a different answer from them.

It's only impossible in the way that multiplying 3*3 and getting 10 is impossible. As in it's impossible to have it get the answer and be doing correct math.

 

[beero1000]

So I understand math, except in a thread titled "Paradox!" in which I claim to show a paradox that was constructed to show how people's intuition about infinity yields mutually contradictory answers. You think I'm showing poor math because I'm showing that there are weird answers that are completely counterintuitive, and seem impossible, when that is the entire point of the paradox - they do seem impossible, but they are just as logically consistent as the intuitive answer.

The reason that the answers are contradictory is because the set you're using to describe the problem isn't the same as the thing you're saying is getting put into the set. It's not a paradox, it's an error. You're taking an equation which doesn't complete and creating a set from after it's completed. You're taking an equation which is using finite operations and then calling it a paradox when infinite operations give a different answer from them.

It's only impossible in the way that multiplying 3*3 and getting 10 is impossible. As in it's impossible to have it get the answer and be doing correct math.

Nonsense.

Using the intuitive labeling, at \(\frac{1}{2^k}\) hours before noon, the labels of the balls in the vase are given by the set \(\{1,2,3,\dots,10k\}\setminus\{10,20,\dots,10k\}\). Using the consecutive labeling, the labels of the balls in the vase are given by the set \(\{1,2,3,\dots,10k\}\setminus\{1,2,3,\dots,k\}\). For every finite \(k\), these two sets have the same cardinality, \(\|\{1,2,3,\dots,10k\}\setminus\{10,20,\dots,10k\}\| = \|\{1,2,3,\dots,10k\}\setminus\{1,2,3,\dots,k\}\|\). For a finite number of operations, it does not matter which balls are removed, or in what order. This matches peoples' intuitions, and they expect that in the limit the same result holds at noon.

However, in the limit, at the time \(\lim_{k\to\infty} 12 - \frac{1}{2^k} = 12\) (exactly) the two sets have different cardinalities. \(\lim_{k\to\infty} \|\{1,2,3,\dots,10k\}\setminus\{10,20,\dots,10k\}\| = \aleph_0\), while \(\lim_{k\to\infty} \|\{1,2,3,\dots,10k\}\setminus\{1,2,\dots,k\}\| = 0\). For infinite sets, it is not sufficient to simply say "add 10 and remove 1" because that is not enough information to specify the limit.

There is nothing wrong with the mathematics. Both are right, and your intuition is faulty. That is the paradox.

 

[Tom Sawyer]

Ya, that's my point. You randomly added infinity to the second set of equations there. If I were multiplying a series of numbers and randomly added a zero into the middle of it, the result would be zero but that wouldn't say anything about the series of number before I went ahead and did that.

You're saying I have this finite thing here and this different infinite thing over there and look! They're different!

...
...
...

Ta da!

 

[Archimedes]

What step of ball manipulation is he performing when it becomes Noon?

What step of addition (0.9 + 0.09 + 0.009 + 0.009 + ....) is being performed when 0.999999... Becomes 1.0000000?

Or do you want to start arguing that 0.9999999999999... Isn't the same thing as 1.0000000?

 

[steve_bnk]

Is anyone really arguing 0.999999 is actually 1? It aint.

'….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: Atnoon, how many balls are in the vase?.


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


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

1. 0 + 10 -1 = 9
2. 9 + 10 -1 = 18
3. 18 + 10 - 1 = 27
4. 27 + 10 -1 = 36
5. 36 + 10 -1 = 45

I do not see how you get 42 in the vase regardless.


I don't see how a limit answers the problem. The limit is taken at affinity But for any finite steps theremaining distance islways > zero.


For Zeno's paradox the exponential equation is


Remaining Distance = d – d * [e^(-n/k) ] where d is total distance, k is solved for one n andremaining distance .and n is 1,2,3...


For any finite step n, the remaining distance is always > zero. In the limit as as n -> inf remaining distance goes to zero but it never really gets there in any finite countable sense..

beero, show me where my simulation is in error...the following algorithm executes your problem as stated.


// scilab

clear;

//One hour = 3600 seconds so thestarting hour is irrelevant.


_time = 0; // time counter
time_remaining = 3600; // initial time remaining
vase = 0; // ball counter
t_stop = 3600; // end point


for i = 1:30;
vase = (vase + 10) -1;
_time = _time + (time_remaining/2);
time_remaining = t_stop - _time;
tt(i,3) = _time;
tt(i,1) = i;
tt(i,2) = vase;
t(i) = _time;
end;

plot2d(t);


tt




Iteration – vase count - time
1. 9. 1800.
2. 18. 2700.
3. 27. 3150.
4. 36. 3375.
5. 45. 3487.5
6. 54. 3543.75
7. 63. 3571.875
8. 72. 3585.9375
9. 81. 3592.9688
10. 90. 3596.4844
11. 99. 3598.2422
12. 108. 3599.1211
13. 117. 3599.5605
14. 126. 3599.7803
15. 135. 3599.8901
16. 144. 3599.9451
17. 153. 3599.9725
18. 162. 3599.9863
19. 171. 3599.9931
20. 180. 3599.9966
21. 189. 3599.9983
22. 198. 3599.9991
23. 207. 3599.9996
24. 216. 3599.9998
25. 225. 3599.9999
26. 234. 3599.9999
27. 243. 3600.
28. 252. 3600.
29. 261. 3600.
30. 270. 3600.

 

[Juma]

Is anyone really arguing 0.999999 is actually 1? It aint.
.

No but 0.9999... (Where ... indicates a never ending repetition) = 1

 

[Underseer]

Is anyone really arguing 0.999999 is actually 1? It aint.

[....]

Oh look, someone else who doesn't understand what infinity is and thinks infinity is just a really large finite number.