Paradox!

[Jimmy Higgins]

Then can you specify a ball that remains?

No. But since the influx is greater than the out flux there is a net inflow of balls keeping count> 0.

Do you also say that ever river is empty since no water molecule stays?

There is a net inflow of balls for every time strictly before noon, but not enough information has been given to determine a specific value at noon.

That is because Noon never happens. You ask for a number based at a point in time, that based on the conditions you set, never happens.

If one can show that every molecule that has ever entered the river is gone at noon, then the river is empty at noon.

Odd metaphor. Your hypothetical puts a dam in the river, that only lets out 10% of the inflow, and for some reason you want to say the river is empty by noon? Every half time, the water increases 1 inch, but is then immediately dropped 0.1 inches. You seem to think this leads to an empty river.

 

[beero1000]

Then can you specify a ball that remains?

No. But since the influx is greater than the out flux there is a net inflow of balls keeping count> 0.

Do you also say that ever river is empty since no water molecule stays?

There is a net inflow of balls for every time strictly before noon, but not enough information has been given to determine a specific value at noon.

That is because Noon never happens. You ask for a number based at a point in time, that based on the conditions you set, never happens.

If one can show that every molecule that has ever entered the river is gone at noon, then the river is empty at noon.

Odd metaphor. Your hypothetical puts a dam in the river, that only lets out 10% of the inflow, and for some reason you want to say the river is empty by noon? Every half time, the water increases 1 inch, but is then immediately dropped 0.1 inches. You seem to think this leads to an empty river.

Zeno's paradoxes have been satisfactorily resolved for centuries. Unless you're a strict finitist and disallow any type of infinity, noon definitely does happen.


I've purposely tried to avoid formalism in this thread, but this is the logic board and the sloppy thinking here is starting to bum me out. The 'intuitive' labeling implies an infinite number of balls remain because \(\|\mathbb{N} \setminus \bigcup_{i=1}^\infty \{10i\}\| = \aleph_{0}\), while another labeling implies no balls remain because \(\|\mathbb{N} \setminus \bigcup_{i=1}^\infty \{i\}\|= 0\).

The counterintuitive nature of the paradox is derived from people intuitively extending the natural and correct finite case to the infinite case, where it no longer works. That is, for any finite numbers \(N,n\) with \(N > 10n\), it is true that \(\|\{1,2,\dots,N\} \setminus \bigcup_{i=1}^n \{i\}\| = \|\{1,2,\dots,N\} \setminus \bigcup_{i=1}^n\{10i\}\|\), but the result is not true if we replace the finite sets with infinite sets.

 

[Tom Sawyer]

It never equals 12:00, it only almost equals 12:00. Why are those infitesimally small fractions of a second a more relevant span of time than the 15 minutes you have at 11:45?

 

[beero1000]

It never equals 12:00, it only almost equals 12:00. Why are those infitesimally small fractions of a second a more relevant span of time than the 15 minutes you have at 11:45?

That is incorrect. The infinite sum exactly equals 12:00. It is a concrete mathematical object, not a moving target, and there is only one logically valid value - noon. You'll never get there after a finite number of operations, but that is the point of the paradox - that an infinite number of operations acts in a very weird way, and can give results different than finite numbers of operations.

 

[ryan]

It never equals 12:00, it only almost equals 12:00. Why are those infitesimally small fractions of a second a more relevant span of time than the 15 minutes you have at 11:45?

The 15 minutes still go by, and then on to 12:01.

 

[Tom Sawyer]

It never equals 12:00, it only almost equals 12:00. Why are those infitesimally small fractions of a second a more relevant span of time than the 15 minutes you have at 11:45?

The 15 minutes still go by, and then on to 12:01.

I don't think you're following the thread.

 

[Tom Sawyer]

It never equals 12:00, it only almost equals 12:00. Why are those infitesimally small fractions of a second a more relevant span of time than the 15 minutes you have at 11:45?

That is incorrect. The infinite sum exactly equals 12:00. It is a concrete mathematical object, not a moving target, and there is only one logically valid value - noon. You'll never get there after a finite number of operations, but that is the point of the paradox - that an infinite number of operations acts in a very weird way, and can give results different than finite numbers of operations.

But they're doing the operation a finite amount of times. Every time they change balls, they add one to the finite number of times they've already done that. There is no finite number which equals infinity when you add one to it.

 

[Deepak]

It never equals 12:00, it only almost equals 12:00. Why are those infitesimally small fractions of a second a more relevant span of time than the 15 minutes you have at 11:45?

That is incorrect. The infinite sum exactly equals 12:00. It is a concrete mathematical object, not a moving target, and there is only one logically valid value - noon. You'll never get there after a finite number of operations, but that is the point of the paradox - that an infinite number of operations acts in a very weird way, and can give results different than finite numbers of operations.

A priest and a rabbi walk into a deli. The priest says "hey what do you call an infinite set that doesn't contain little omega", and the rabbi says "I'll have the brisket"

 

[beero1000]

It never equals 12:00, it only almost equals 12:00. Why are those infitesimally small fractions of a second a more relevant span of time than the 15 minutes you have at 11:45?

The 15 minutes still go by, and then on to 12:01.

I don't think you're following the thread.

He's doing better than most.

It never equals 12:00, it only almost equals 12:00. Why are those infitesimally small fractions of a second a more relevant span of time than the 15 minutes you have at 11:45?

That is incorrect. The infinite sum exactly equals 12:00. It is a concrete mathematical object, not a moving target, and there is only one logically valid value - noon. You'll never get there after a finite number of operations, but that is the point of the paradox - that an infinite number of operations acts in a very weird way, and can give results different than finite numbers of operations.

But they're doing the operation a finite amount of times. Every time they change balls, they add one to the finite number of times they've already done that. There is no finite number which equals infinity when you add one to it.

At noon, they have done more than any finite number of operations.

It never equals 12:00, it only almost equals 12:00. Why are those infitesimally small fractions of a second a more relevant span of time than the 15 minutes you have at 11:45?

That is incorrect. The infinite sum exactly equals 12:00. It is a concrete mathematical object, not a moving target, and there is only one logically valid value - noon. You'll never get there after a finite number of operations, but that is the point of the paradox - that an infinite number of operations acts in a very weird way, and can give results different than finite numbers of operations.

A priest and a rabbi walk into a deli. The priest says "hey what do you call an infinite set that doesn't contain little omega", and the rabbi says "I'll have the brisket"

A rabbi walks into a bar and sees a priest sitting there saying "... 9, 5, 1, 4, 1, 3". The rabbi says "hey, what are you doing?", and the priest says "Oh, I just finished reciting all the digits of pi, backwards."

 

[Tom Sawyer]

It never equals 12:00, it only almost equals 12:00. Why are those infitesimally small fractions of a second a more relevant span of time than the 15 minutes you have at 11:45?

That is incorrect. The infinite sum exactly equals 12:00. It is a concrete mathematical object, not a moving target, and there is only one logically valid value - noon. You'll never get there after a finite number of operations, but that is the point of the paradox - that an infinite number of operations acts in a very weird way, and can give results different than finite numbers of operations.

But they're doing the operation a finite amount of times. Every time they change balls, they add one to the finite number of times they've already done that. There is no finite number which equals infinity when you add one to it.

At noon, they have done more than any finite number of operations.

OK, at what point before noon did they get to that? They did the operation a finite number of times, then did it once more and it became an infinite number of times. What is the finite number you are alleging that they were on before that happened?

 

[beero1000]

It never equals 12:00, it only almost equals 12:00. Why are those infitesimally small fractions of a second a more relevant span of time than the 15 minutes you have at 11:45?

That is incorrect. The infinite sum exactly equals 12:00. It is a concrete mathematical object, not a moving target, and there is only one logically valid value - noon. You'll never get there after a finite number of operations, but that is the point of the paradox - that an infinite number of operations acts in a very weird way, and can give results different than finite numbers of operations.

But they're doing the operation a finite amount of times. Every time they change balls, they add one to the finite number of times they've already done that. There is no finite number which equals infinity when you add one to it.

At noon, they have done more than any finite number of operations.

OK, at what point before noon did they get to that? They did the operation a finite number of times, then did it once more and it became an infinite number of times. What is the finite number you are alleging that they were on before that happened?

Get to what? Give me a finite number and I can tell you exactly when they surpassed that number of operations.

You cannot get an infinite number by adding one to a finite number, but you can get one as a limit of finite numbers, which is what happens in this scenario.

 

[Tom Sawyer]

OK, at what point before noon did they get to that? They did the operation a finite number of times, then did it once more and it became an infinite number of times. What is the finite number you are alleging that they were on before that happened?

Get to what? Give me a finite number and I can tell you exactly when they surpassed that number of operations.

You cannot get an infinite number by adding one to a finite number, but you can get one as a limit of finite numbers, which is what happens in this scenario.

You can't get to an infinite number by adding one to a finite number but there is no limit to finite numbers. You just keep getting to a larger and larger finite number with each iteration and you never reach an end. You don't reach some limit of finite numbers after the process runs on for a while and find yourself at infinity. It's just some larger finite number each time.

 

[Juma]

OK, at what point before noon did they get to that? They did the operation a finite number of times, then did it once more and it became an infinite number of times. What is the finite number you are alleging that they were on before that happened?

Get to what? Give me a finite number and I can tell you exactly when they surpassed that number of operations.

You cannot get an infinite number by adding one to a finite number, but you can get one as a limit of finite numbers, which is what happens in this scenario.

You can't get to an infinite number by adding one to a finite number but there is no limit to finite numbers. You just keep getting to a larger and larger finite number with each iteration and you never reach an end. You don't reach some limit of finite numbers after the process runs on for a while and find yourself at infinity. It's just some larger finite number each time.

Yes? And your problem with this is?

 

[ryan]

OK, at what point before noon did they get to that? They did the operation a finite number of times, then did it once more and it became an infinite number of times. What is the finite number you are alleging that they were on before that happened?

Get to what? Give me a finite number and I can tell you exactly when they surpassed that number of operations.

You cannot get an infinite number by adding one to a finite number, but you can get one as a limit of finite numbers, which is what happens in this scenario.

You can't get to an infinite number by adding one to a finite number but there is no limit to finite numbers. You just keep getting to a larger and larger finite number with each iteration and you never reach an end. You don't reach some limit of finite numbers after the process runs on for a while and find yourself at infinity. It's just some larger finite number each time.

You never get to the last one. A great geometrical representation of this, and how large infinity is in general, is the expanding circle.

Draw a circle. Draw a line through the circle in half so that you have to halves. Now, draw a line through the center of the line perpendicularly so that you have 4 equal quarters. Continue to cut each pie piece in half until you have 8 equal pie pieces. You may notice that every time you cut one pie piece in half you must do twice as many cuts as you did the last time: 2, 4, 8, 16, 32 ....

If you got to 8, then you will see that you now have 8 points on the circumference of the circle. There is an infinite number of points on the circumference, so you must be 8 points closer than you were before right? But wait, every time you make points between the pie pieces, you are actually doubling the number of points that you need to make next time. Are we getting closer to infinity, or are we getting further from infinity?

To make it even worse, imagine drawing a larger circle around your original circle. Now your gaps have got larger if you extend the lines to the larger circumference. The larger circle has the same number of points (the same infinity) as the smaller circle!?!?!? Keep in mind that this is the smallest infinity.

That's how f***ed up infinity is. That's why any attempt at counting infinity or even understanding how large it is, is futile.

 

[Archimedes]

How can people say noon never arrives? The puzzle specifies that it starts at 11am. An hour after that it will be noon.

How can a timespan defined in the puzzle as 1 hour take any more than 1 hour?

 

[steve_bnk]

Then can you specify a ball that remains?

No. But since the influx is greater than the out flux there is a net inflow of balls keeping count> 0.

Do you also say that ever river is empty since no water molecule stays?

There is a net inflow of balls for every time strictly before noon, but not enough information has been given to determine a specific value at noon.

That is because Noon never happens. You ask for a number based at a point in time, that based on the conditions you set, never happens.

If one can show that every molecule that has ever entered the river is gone at noon, then the river is empty at noon.

Odd metaphor. Your hypothetical puts a dam in the river, that only lets out 10% of the inflow, and for some reason you want to say the river is empty by noon? Every half time, the water increases 1 inch, but is then immediately dropped 0.1 inches. You seem to think this leads to an empty river.

Zeno's paradoxes have been satisfactorily resolved for centuries. Unless you're a strict finitist and disallow any type of infinity, noon definitely does happen.


I've purposely tried to avoid formalism in this thread, but this is the logic board and the sloppy thinking here is starting to bum me out. The 'intuitive' labeling implies an infinite number of balls remain because \(\|\mathbb{N} \setminus \bigcup_{i=1}^\infty \{10i\}\| = \aleph_{0}\), while another labeling implies no balls remain because \(\|\mathbb{N} \setminus \bigcup_{i=1}^\infty \{i\}\|= 0\).

The counterintuitive nature of the paradox is derived from people intuitively extending the natural and correct finite case to the infinite case, where it no longer works. That is, for any finite numbers \(N,n\) with \(N > 10n\), it is true that \(\|\{1,2,\dots,N\} \setminus \bigcup_{i=1}^n \{i\}\| = \|\{1,2,\dots,N\} \setminus \bigcup_{i=1}^n\{10i\}\|\), but the result is not true if we replace the finite sets with infinite sets.

One simulation is worth a thousand mathematical obfuscations.

 

[Tom Sawyer]

OK, at what point before noon did they get to that? They did the operation a finite number of times, then did it once more and it became an infinite number of times. What is the finite number you are alleging that they were on before that happened?

Get to what? Give me a finite number and I can tell you exactly when they surpassed that number of operations.

You cannot get an infinite number by adding one to a finite number, but you can get one as a limit of finite numbers, which is what happens in this scenario.

You can't get to an infinite number by adding one to a finite number but there is no limit to finite numbers. You just keep getting to a larger and larger finite number with each iteration and you never reach an end. You don't reach some limit of finite numbers after the process runs on for a while and find yourself at infinity. It's just some larger finite number each time.

Yes? And your problem with this is?

People who say it's wrong and that you can count up to infinity using finite numbers.

 

[Tom Sawyer]

OK, at what point before noon did they get to that? They did the operation a finite number of times, then did it once more and it became an infinite number of times. What is the finite number you are alleging that they were on before that happened?

Get to what? Give me a finite number and I can tell you exactly when they surpassed that number of operations.

You cannot get an infinite number by adding one to a finite number, but you can get one as a limit of finite numbers, which is what happens in this scenario.

You can't get to an infinite number by adding one to a finite number but there is no limit to finite numbers. You just keep getting to a larger and larger finite number with each iteration and you never reach an end. You don't reach some limit of finite numbers after the process runs on for a while and find yourself at infinity. It's just some larger finite number each time.

You never get to the last one. A great geometrical representation of this, and how large infinity is in general, is the expanding circle.

Draw a circle. Draw a line through the circle in half so that you have to halves. Now, draw a line through the center of the line perpendicularly so that you have 4 equal quarters. Continue to cut each pie piece in half until you have 8 equal pie pieces. You may notice that every time you cut one pie piece in half you must do twice as many cuts as you did the last time: 2, 4, 8, 16, 32 ....

If you got to 8, then you will see that you now have 8 points on the circumference of the circle. There is an infinite number of points on the circumference, so you must be 8 points closer than you were before right? But wait, every time you make points between the pie pieces, you are actually doubling the number of points that you need to make next time. Are we getting closer to infinity, or are we getting further from infinity?

To make it even worse, imagine drawing a larger circle around your original circle. Now your gaps have got larger if you extend the lines to the larger circumference. The larger circle has the same number of points (the same infinity) as the smaller circle!?!?!? Keep in mind that this is the smallest infinity.

That's how f***ed up infinity is. That's why any attempt at counting infinity or even understanding how large it is, is futile.

Yes, there are an infinite number of ways to halve a circle. One cannot, however, draw and infinite number of lines through a circle. Once you start drawing lines, each line that you draw is a finite number. You never reach th end of potential lines to draw and you never reach an infinite number of lines.

- - - Updated - - -

How can people say noon never arrives? The puzzle specifies that it starts at 11am. An hour after that it will be noon.

How can a timespan defined in the puzzle as 1 hour take any more than 1 hour?

Well, which interval is it that you feel completes the hour and gets you to noon?

 

[beero1000]

Then can you specify a ball that remains?

No. But since the influx is greater than the out flux there is a net inflow of balls keeping count> 0.

Do you also say that ever river is empty since no water molecule stays?

There is a net inflow of balls for every time strictly before noon, but not enough information has been given to determine a specific value at noon.

That is because Noon never happens. You ask for a number based at a point in time, that based on the conditions you set, never happens.

If one can show that every molecule that has ever entered the river is gone at noon, then the river is empty at noon.

Odd metaphor. Your hypothetical puts a dam in the river, that only lets out 10% of the inflow, and for some reason you want to say the river is empty by noon? Every half time, the water increases 1 inch, but is then immediately dropped 0.1 inches. You seem to think this leads to an empty river.

Zeno's paradoxes have been satisfactorily resolved for centuries. Unless you're a strict finitist and disallow any type of infinity, noon definitely does happen.


I've purposely tried to avoid formalism in this thread, but this is the logic board and the sloppy thinking here is starting to bum me out. The 'intuitive' labeling implies an infinite number of balls remain because \(\|\mathbb{N} \setminus \bigcup_{i=1}^\infty \{10i\}\| = \aleph_{0}\), while another labeling implies no balls remain because \(\|\mathbb{N} \setminus \bigcup_{i=1}^\infty \{i\}\|= 0\).

The counterintuitive nature of the paradox is derived from people intuitively extending the natural and correct finite case to the infinite case, where it no longer works. That is, for any finite numbers \(N,n\) with \(N > 10n\), it is true that \(\|\{1,2,\dots,N\} \setminus \bigcup_{i=1}^n \{i\}\| = \|\{1,2,\dots,N\} \setminus \bigcup_{i=1}^n\{10i\}\|\), but the result is not true if we replace the finite sets with infinite sets.

One simulation is worth a thousand mathematical obfuscations.

And a simulation of the wrong thing is worse than worthless.

OK, at what point before noon did they get to that? They did the operation a finite number of times, then did it once more and it became an infinite number of times. What is the finite number you are alleging that they were on before that happened?

Get to what? Give me a finite number and I can tell you exactly when they surpassed that number of operations.

You cannot get an infinite number by adding one to a finite number, but you can get one as a limit of finite numbers, which is what happens in this scenario.

You can't get to an infinite number by adding one to a finite number but there is no limit to finite numbers. You just keep getting to a larger and larger finite number with each iteration and you never reach an end. You don't reach some limit of finite numbers after the process runs on for a while and find yourself at infinity. It's just some larger finite number each time.

Yes? And your problem with this is?

People who say it's wrong and that you can count up to infinity using finite numbers.

Do you not understand the concept of limit? No one is saying you switch from finite numbers to an infinite number while counting, but the limit of counting numbers is infinite.

 

[Jimmy Higgins]

Then can you specify a ball that remains?

No. But since the influx is greater than the out flux there is a net inflow of balls keeping count> 0.

Do you also say that ever river is empty since no water molecule stays?

There is a net inflow of balls for every time strictly before noon, but not enough information has been given to determine a specific value at noon.

That is because Noon never happens. You ask for a number based at a point in time, that based on the conditions you set, never happens.

If one can show that every molecule that has ever entered the river is gone at noon, then the river is empty at noon.

Odd metaphor. Your hypothetical puts a dam in the river, that only lets out 10% of the inflow, and for some reason you want to say the river is empty by noon? Every half time, the water increases 1 inch, but is then immediately dropped 0.1 inches. You seem to think this leads to an empty river.

I've purposely tried to avoid formalism in this thread, but this is the logic board and the sloppy thinking here is starting to bum me out.

Shall we lift your pedestal higher?

The 'intuitive' labeling implies an infinite number of balls remain because \(\|\mathbb{N} \setminus \bigcup_{i=1}^\infty \{10i\}\| = \aleph_{0}\), while another labeling implies no balls remain because \(\|\mathbb{N} \setminus \bigcup_{i=1}^\infty \{i\}\|= 0\).

And the other may suggest that if a finite amount of time has passed, there be balls in the jar. Your question parallel to a man walking towards the finish line is that given an infinite number of steps he never actually moved an inch.

The counterintuitive nature of the paradox is derived from people intuitively extending the natural and correct finite case to the infinite case, where it no longer works. That is, for any finite numbers \(N,n\) with \(N > 10n\), it is true that \(\|\{1,2,\dots,N\} \setminus \bigcup_{i=1}^n \{i\}\| = \|\{1,2,\dots,N\} \setminus \bigcup_{i=1}^n\{10i\}\|\), but the result is not true if we replace the finite sets with infinite sets.

Could you enlighten us to let us know when the number of balls in the jar starts decreasing. Please keep in mind that the act of adding and removing the balls is at the same time, according to your paradox. Therefore, you are really only adding 9 balls at a time, not adding one in one step and removing one in another.