Infinity Paradoxes - Numberphile
video description
Hilberts Hotel: the paradox merely arises from semantics and potential misnomer. If a hotel has infinite rooms, it is impossible to fill every room. If every room is filled, then there are not an infinite number of rooms.
Gabriel's Trumpet: another false assumption. Saying the trumpet thing has infinite area but finite volume is like saying a sphere has infinite area (it comes to an infinite number of points, not just one) and finite volume. Things in our universe can't get below the planck length anyway. This is just a musical instrument version of the Achilles and the tortoise paradox, but measuring geometry of a shape rather than distance.
Dartboard Paradox: It has a greater than 0 chance of hitting the dartboard, and even though there may be infinitely many points on the dartboard mathematically (physically there isn't due to the size of the dart tip compared to the size of the dart board and taking into account the planck length, there is nothing to say that all of those infinitesimal -greater than 0- chances add up to or tend toward infinity. To simplify: if there are infinity -points- you can hit on the dart board, any specific point has a 1 over infinity chance of being hit (assuming all darts hit the board. You add up all of the 1 over infinity chances, and you get infinity over infinity, which isn't an -infinity number of chances-, it's 1. As in to hit the board its inf/inf or 1 chance, and to hit any specific point on the board its a 1/infinity chance. There is no paradox here.
Double your money paradox: The real expected value isn't infinite because there isn't infinite money in the world. Also, the amount of money you spend on the game would most likely tend toward infinity also. Since the odds of the game ending at any stage are 50/50 (assuming a fair coin, most games would be expected to end within 10 stages, surely. How often does someone flip a coin and get 10 heads or 10 tails in a row? So betting anything more than the payout for 10 in a row would be absurd, let alone betting your house or life savings.
Date: 2022-04-08
Related videos
• TED-Ed
Comments and reviews: 9
Peter
INFINITY = 1 sphere where the smaller and larger are main attribute of IT and WHY it is a SOLID and = 1. please make sure I get my Nobel prize as i alone am correct o0n all the planet and all you math guys could not count to see it and all i had to do is look and see it and see all of you wrong same time. Infinity is all sizes of spheres as 1 sphere and so HAS NO MOVING PARTS. IT JUST IS.
ETERNITY is ALSO a SPHERE and also smaller and larger same time and also has NO MOVING PARTS. time does not work in 1 of ALL times as 1. one has to look inside eternity to see a LINE same as infinity. you pick lines inwards or outwards in layers we give them but they have no layers as all of it as 1. So paradoxes? time and all that sets stuff. LOL sorry but NO!
just 2 spheres and you can even put in exact same core spot and no effect. still 1 sphere of infinity and eternity = 1 as same thing same sphere. we just see the sphere differently ONLY and so it always = 1 always! Please if you see I AM CORRECT and whole planet is wrong, Give me my credit. I KNOW it is IMPOSSIBLE to prove me wrong. i know what I can see and i am correct and a sea of math is wrong and needs tweaking now to -lines of infinity-. and YES you can pick a OFF core line. and HOW you make a shape exist in infinity does have effect. a spiral infinite LINE in INFINITY CAN change size as enters the core and exit it over and over endlessly in all sizes. other MATH SHAPES also can have such enter and exist -core of infinity- [ My term? ] and out and still be in 1 and adding a form in infinity IS possible with no paradox. a wavy line also as example can enter and exit the core infinity size never ending small and BACK OUT to large set size repeating it endlessly as a skip from core of infinite smaller to balanced set size always smaller into the larger layers. or same size as relative. some cools stuff NO PARADOXES! Nobel please LOL.
reply
INFINITY = 1 sphere where the smaller and larger are main attribute of IT and WHY it is a SOLID and = 1. please make sure I get my Nobel prize as i alone am correct o0n all the planet and all you math guys could not count to see it and all i had to do is look and see it and see all of you wrong same time. Infinity is all sizes of spheres as 1 sphere and so HAS NO MOVING PARTS. IT JUST IS.
ETERNITY is ALSO a SPHERE and also smaller and larger same time and also has NO MOVING PARTS. time does not work in 1 of ALL times as 1. one has to look inside eternity to see a LINE same as infinity. you pick lines inwards or outwards in layers we give them but they have no layers as all of it as 1. So paradoxes? time and all that sets stuff. LOL sorry but NO!
just 2 spheres and you can even put in exact same core spot and no effect. still 1 sphere of infinity and eternity = 1 as same thing same sphere. we just see the sphere differently ONLY and so it always = 1 always! Please if you see I AM CORRECT and whole planet is wrong, Give me my credit. I KNOW it is IMPOSSIBLE to prove me wrong. i know what I can see and i am correct and a sea of math is wrong and needs tweaking now to -lines of infinity-. and YES you can pick a OFF core line. and HOW you make a shape exist in infinity does have effect. a spiral infinite LINE in INFINITY CAN change size as enters the core and exit it over and over endlessly in all sizes. other MATH SHAPES also can have such enter and exist -core of infinity- [ My term? ] and out and still be in 1 and adding a form in infinity IS possible with no paradox. a wavy line also as example can enter and exit the core infinity size never ending small and BACK OUT to large set size repeating it endlessly as a skip from core of infinite smaller to balanced set size always smaller into the larger layers. or same size as relative. some cools stuff NO PARADOXES! Nobel please LOL.
reply
johanh
I'm questioning the whole idea of a paradox with Hilbert's Hotel as presented here. The whole concept is based on that you can have two or more persons per room doing the shifting operation. Of course it works then, when you can keep shifting them towards infinity. But there is always a person in the next room that will be kicked into next room. If you add more people to the hotel, basically you are adding more pairs shifting all the time towards infinity. Even if you start by adding just one guest, you have created an infinite shifting operation that never stops (you basically started an infinite musical chair game. If you add an infinite amount of people, eventually you end up with a whole chain of shifting operations, where at any moment, there are always two or more people doing the shifting operation for each room. If you allow more than two persons per room doing the shifting, eventually you end up with an infinite amount of persons for each room doing the shifting operation. You can complicate matters even more, if you say that the shifting operation can be done infinitely fast. The whole thing becomes kind of silly, but I don't really see where the paradox is. Maybe I missed the whole point!
reply
I'm questioning the whole idea of a paradox with Hilbert's Hotel as presented here. The whole concept is based on that you can have two or more persons per room doing the shifting operation. Of course it works then, when you can keep shifting them towards infinity. But there is always a person in the next room that will be kicked into next room. If you add more people to the hotel, basically you are adding more pairs shifting all the time towards infinity. Even if you start by adding just one guest, you have created an infinite shifting operation that never stops (you basically started an infinite musical chair game. If you add an infinite amount of people, eventually you end up with a whole chain of shifting operations, where at any moment, there are always two or more people doing the shifting operation for each room. If you allow more than two persons per room doing the shifting, eventually you end up with an infinite amount of persons for each room doing the shifting operation. You can complicate matters even more, if you say that the shifting operation can be done infinitely fast. The whole thing becomes kind of silly, but I don't really see where the paradox is. Maybe I missed the whole point!
reply
frame-perfect
Breaking the dark up in 2 areas doesn't solve the problem because those areas can overlap. If there is no limit to how slightly you can adjust the dart, then there are an infinite number of positions. No matter how slightly you adjust the dark, you could have adjusted it by half that amount because the possible positions overlap. You don't need to introduce the concept of a point to make it paradoxical, and going by a finite area doesn't resolve the paradox.
The reason the concept of infinity produces paradoxes is that we live in a finite world where there is a limitation of greatness and smallness. There is a limitation to how slightly you can adjust the dart.
reply
Breaking the dark up in 2 areas doesn't solve the problem because those areas can overlap. If there is no limit to how slightly you can adjust the dart, then there are an infinite number of positions. No matter how slightly you adjust the dark, you could have adjusted it by half that amount because the possible positions overlap. You don't need to introduce the concept of a point to make it paradoxical, and going by a finite area doesn't resolve the paradox.
The reason the concept of infinity produces paradoxes is that we live in a finite world where there is a limitation of greatness and smallness. There is a limitation to how slightly you can adjust the dart.
reply
Clint
The Hilbert hotel paradox is flawed. To move the guests and have one guest per room, you would have to start moving the people from the side of infinity first so that the room would actually get empty for the new guest to occupy. The guest would be waiting an infinite amount of time for room number 1 to be occupied. Or, there would always be 2 guests in one room at all times. OR, there would always be a guest stuck between 2 occupied rooms. So, a hotel with an infinite number of rooms would not be able to accept any new guests. It would always have the same amount of rooms occupied by the same amount of guests. Or rooms with multiple guests.
reply
The Hilbert hotel paradox is flawed. To move the guests and have one guest per room, you would have to start moving the people from the side of infinity first so that the room would actually get empty for the new guest to occupy. The guest would be waiting an infinite amount of time for room number 1 to be occupied. Or, there would always be 2 guests in one room at all times. OR, there would always be a guest stuck between 2 occupied rooms. So, a hotel with an infinite number of rooms would not be able to accept any new guests. It would always have the same amount of rooms occupied by the same amount of guests. Or rooms with multiple guests.
reply
Prashanta
On the dartboard problem, here's my two cents. First, when you think of the dart hitting a point instead of an area, it becomes a probability density function. The value of the probability density function at any single point is zero. So the answer to the question of the probability of hitting the dart at my point should, in fact, be zero. Now the idea that if you sum the probability of hitting each point becomes zero is wrong. Since there are infinite points, the total probability is Zero-Infinity. Zero-Infinity is not zero, it is indeterminate. Indeterminates can have values in specific cases. In this case, the value is one.
reply
On the dartboard problem, here's my two cents. First, when you think of the dart hitting a point instead of an area, it becomes a probability density function. The value of the probability density function at any single point is zero. So the answer to the question of the probability of hitting the dart at my point should, in fact, be zero. Now the idea that if you sum the probability of hitting each point becomes zero is wrong. Since there are infinite points, the total probability is Zero-Infinity. Zero-Infinity is not zero, it is indeterminate. Indeterminates can have values in specific cases. In this case, the value is one.
reply
Noel
Hilberts Hotel destroys Cantors second diagonal argument. As you go down diagonally adding -1- according to his algorithm, you would find yourself not convicting the list of natural numbers of its incompleteness towards their irrational counterparts, but rather discovering its completeness, as with every new number on the diagonal line, you discover a new number, already being linked somewhere in the list beneath the index. You would never finish, you would just see the numbers randomly lighting up somewhere down the endless list. Because infinity is faster than you, as it was there all along. This is truly beautiful.
reply
Hilberts Hotel destroys Cantors second diagonal argument. As you go down diagonally adding -1- according to his algorithm, you would find yourself not convicting the list of natural numbers of its incompleteness towards their irrational counterparts, but rather discovering its completeness, as with every new number on the diagonal line, you discover a new number, already being linked somewhere in the list beneath the index. You would never finish, you would just see the numbers randomly lighting up somewhere down the endless list. Because infinity is faster than you, as it was there all along. This is truly beautiful.
reply
---
Well, I thought, that on the trumpet paradox, even if you pour a bucket of paint in it, it would filter down and run out at some point. We would need to set how thick the paint is going to cover the surface. If the paint will cover the surface infinitely thin, yes, pouring it down, the paint would go down infinitely. But then you can also cover the surface with a paintbrush. I want to say that a infinitely small number is zero, but if any of you have better knowledge of infinity, and don't think so, please inform me. Anyway saying that an infinitely small number is zero, you can't run out.
reply
Well, I thought, that on the trumpet paradox, even if you pour a bucket of paint in it, it would filter down and run out at some point. We would need to set how thick the paint is going to cover the surface. If the paint will cover the surface infinitely thin, yes, pouring it down, the paint would go down infinitely. But then you can also cover the surface with a paintbrush. I want to say that a infinitely small number is zero, but if any of you have better knowledge of infinity, and don't think so, please inform me. Anyway saying that an infinitely small number is zero, you can't run out.
reply
Edward
1. The hotel is defined as being full and having infinite rooms. In other words, it has an infinite number of full rooms. You can't move the guests along because there will only be full rooms in front. -
2. If the trumpet has infinite length, it has infinite volume and infinite surface area. You would need an infinite amount of paint. -
3. If it's a nomal dart, it will hit an infinite number of points. The odds would be reasonable. If the dart has infinitesimal cross sectional area, it has infinitesimal odds of hitting a chosen point. -
4. Ok, it's not really a paradox though.
reply
1. The hotel is defined as being full and having infinite rooms. In other words, it has an infinite number of full rooms. You can't move the guests along because there will only be full rooms in front. -
2. If the trumpet has infinite length, it has infinite volume and infinite surface area. You would need an infinite amount of paint. -
3. If it's a nomal dart, it will hit an infinite number of points. The odds would be reasonable. If the dart has infinitesimal cross sectional area, it has infinitesimal odds of hitting a chosen point. -
4. Ok, it's not really a paradox though.
reply
Mounted
I'm confused about Hilbert's Hotel. Practically, what it's saying is that you can superimpose one infinity over top of another infinity. And then you can make the top infinity bigger while keeping the other infinity the same size, and they will still be perfectly superimposed? In the animation, the last room drawn on the paper had two people in it, so does that suggest that one room has to end up having two guests in it together, but since the rooms are infinite, it will never come about that any one room will end up permanently with two guests? Am I misunderstanding something?
reply
I'm confused about Hilbert's Hotel. Practically, what it's saying is that you can superimpose one infinity over top of another infinity. And then you can make the top infinity bigger while keeping the other infinity the same size, and they will still be perfectly superimposed? In the animation, the last room drawn on the paper had two people in it, so does that suggest that one room has to end up having two guests in it together, but since the rooms are infinite, it will never come about that any one room will end up permanently with two guests? Am I misunderstanding something?
reply
Add a review, comment
Other channel videos
