Imagine there's a hotel with infinite rooms. They're numbered one, two, three, four, and so on forever. This is the Hilbert Hotel, and you are the manager. Now, it might seem like you could accommodate anyone who ever shows up, but there is a limit, a way to exceed even the infinity of rooms at the Hilbert Hotel. To start, let's say only one person is allowed in each room, and all the rooms are full. There are an infinite number of people in an infinite number of rooms. Then, someone new shows up and they want a room, but all the rooms are occupied. So, what should you do? Well, a lesser manager might turn them away, but you know about infinity. So, you get on the PA and you tell all the guests to move down a room. So the person in room one moves to room two, the one in room two moves to room three, and so on down the line. And now you can put the new guest in room one. If a bus shows up with a hundred people, you know exactly what to do. Just move everyone down a hundred rooms and put the new guests in their vacated rooms. But now, say a bus shows up that is infinitely long, and it's carrying infinitely many people. You knew what to do with a finite number of people, but what do you do with infinite people? You think about it for a minute, and then come up with a plan. You tell each of your existing guests to move to the room with double their room number. So the person in room one moves to room two, room two moves to room four, room three to room six, and so on. And now, all of the odd-numbered rooms are available. And you know there are an infinite number of odd numbers, so you can give each person on the infinite bus a unique, odd-numbered room. This hotel is really starting to feel like it can fit everybody, and that's the beauty of infinity. It goes on forever. And then, all of a sudden, more infinite buses show up. Not just one or two, but an infinite number of infinite buses. So, what can you do? Well, you pull out an infinite spreadsheet, of course. You make a row for each bus, bus one, bus two, bus three, and so on, and a row at the top for all the people who are already in the hotel. The columns are for the position each person occupies. So you've got hotel room one, hotel room two, hotel room three, etc. And then bus one seat one, bus one seat two, bus one seat three, and so on. So each person gets a unique identifier, which is a combination of their vehicle and their position in it. So, how do you assign the rooms? We'll start in the top left corner and draw a line that zigzags back and forth across the spreadsheet, going over each unique ID exactly once. Then, imagine you pull on the opposite ends of this line, straightening it out. So, we've gone from an infinite by infinite grid to a single infinite line. It's then pretty simple just to line up each person on that line with a unique room in the hotel. So, everyone fits, no problem. But now, a big bus pulls up, an infinite party bus with no seats. Instead, everyone on board is identified by their unique name, which is kind of strange. So, their names all consist of only two letters, A and B, but each name is infinitely long. So, someone is named A B B A A A A A A A and so on forever. Someone else is named A B A B A B A B A B, et cetera. On this bus, there's a person with every possible infinite sequence of these two letters. Now, A B B A A A A A, I'll call him Abba for short. He comes into the hotel to arrange the rooms. But you tell him, sorry, there's no way we can fit all of you in the hotel. And he's like, what do you mean? There's an infinite number of us and you have an infinite number of rooms. Why won't this work? So, you show him. You pull out your infinite spreadsheet again and start assigning rooms to people on the bus. So you have room one, assign it to Abba. And then room two to A B A B A B repeating. And you keep going, putting a different string of A's and B's beside each room number. Now, here's the problem, you tell Abba. Let's say we have a complete infinite list. I can still write down the name of a person who doesn't yet have a room. The way you do it is you take the first letter of the first name and flip it from an A to a B. Then take the second letter of the second name and flip it from a B to an A. And you keep doing this all the way down the list. And the name you write down is guaranteed to appear nowhere on that list because it won't match the first letter of the first name, or the second letter of the second name, or the third letter of the third name. It will be different from every name on the list by at least one character, the letter on the diagonal. The number of rooms in the Hilbert Hotel is infinite, sure, but it is countably infinite, meaning there are as many rooms as there are positive integers, one to infinity. By contrast, the number of people on the bus is uncountably infinite. If you try to match up each one with an integer, you will still have people left over. Some infinities are bigger than others. So there's a limit to the people that you can fit in the Hilbert Hotel. This is mind-blowing enough, but what's even crazier is that the discovery of different sized infinities sparked a line of inquiry that led directly to the invention of the device you're watching this on right now. But that's a story for another time.
The camera pushes in on a stylized, slightly run-down hotel on a city street at dusk.
"Imagine there's a hotel with infinite rooms."
Setting: A stylized urban street — Dim, ambient light from the setting sun and streetlights.
Text: "Hilbert Hotel", "NO VACANCY"
The camera moves down a seemingly endless hotel hallway with numbered doors stretching into the distance.
"They're numbered one, two, three, four, and so on forever."
Setting: An infinite hotel hallway — Warm, artificial light from wall sconces.
Text: "1, 2, 3, 4, 5, 6, 7, 8..."
A shot of the hotel reception desk. A large, furry, blue blob-like creature sits behind the desk, representing the manager.
"This is the Hilbert Hotel, and you are the manager."
Setting: Hotel lobby — Overhead spotlights casting a warm glow.
People (1):
• sitting, The entire body is covered in blue fur. hair — Two small, black, beady eyes on a furry blue blob
The camera rapidly moves down a different, warmly lit infinite hallway, creating a motion blur effect.
"a way to exceed even the infinity of rooms at the Hilbert Hotel."
Setting: An infinite hotel hallway — Bright, warm lights creating a blur.
A green, furry blob-person walks up to a door labeled '9', which opens to a white void, and the person disappears inside.
"There are an infinite number of people in an infinite number of rooms."
Setting: Hotel hallway — Warm overhead lighting.
People (1):
• walking, Covered in green fur. hair — Furry green blob with small eyes
Text: "8, 9, 10"
A new, rocky, purple-brown blob-person walks up to the reception desk where the blue manager is.
"Then, someone new shows up and they want a room,"
Setting: Hotel lobby — Overhead spotlights.
People (2):
• walking towards desk, none hair — Rocky, purplish-brown creature
• sitting, Covered in blue fur. hair — Furry blue blob with small eyes
The blue manager blob leans towards a futuristic-looking microphone on the desk.
"you get on the PA and you tell all the guests to move down a room."
Setting: Hotel lobby — Soft, focused light.
People (1):
• leaning forward, Covered in blue fur. hair — Furry blue blob with small eyes
An animation shows colorful blob-people exiting their rooms and moving one door to the right, freeing up Room 1.
"So the person in room one moves to room two, the one in room two moves to room three, and so on down the line."
Setting: Diagrammatic space — Self-illuminated graphics
People (1):
• moving from one door to the next, furry texture hair — Various colored blobs
Text: "1, 2, 3, 4, 5..."
A simple, stylized bus pulls up in front of the Hilbert Hotel.
"If a bus shows up with a hundred people, you know exactly what to do."
Setting: Urban street — Streetlights and hotel lights.
Text: "Hilbert Hotel"
A split-screen shows a red blob-person leaving a doorway labeled '1' on the left, and a blue blob-person entering a doorway labeled '101' on the right.
"Just move everyone down a hundred rooms and put the new guests in their vacated rooms."
Setting: Diagrammatic space — Soft spotlights on the numbers and doorways.
People (2):
• exiting, Red fur hair — Red furry blob
• entering, Blue fur hair — Blue furry blob
Text: "1", "101"
A very long, stylized blue and yellow bus scrolls into the frame from the right, stretching across the entire screen in front of the hotel.
"But now say a bus shows up that is infinitely long, and it's carrying infinitely many people."
Setting: Urban street — Warm streetlights illuminating the pavement.
Text: "Hilbert Hotel"
The blue manager blob leans towards the microphone again.
"You tell each of your existing guests to move to the room with double their room number."
Setting: Hotel lobby — Soft, focused light.
People (1):
• leaning forward, Covered in blue fur. hair — Furry blue blob with small eyes
An animation shows large white arrows indicating guests moving from their current room (n) to a new room (2n).
"So the person in room one moves to room two, room two moves to room four, room three to room six, and so on."
Setting: Diagrammatic space — Self-illuminated graphics
Text: "1, 2, 3, 4, 5..."
The camera pans up along an infinite hotel hallway with red doors and yellow lights.
"and that's the beauty of infinity. It goes on forever."
Setting: Infinite hotel hallway — Warm yellow light from wall sconces.
Text: "237, 238, 239, 240..."
Two simple white buses with yellow headlights pull up to the hotel from opposite sides.
"And then, all of a sudden, more infinite buses show up."
Setting: Urban street at night — Dramatic spotlights and headlights, creating a foggy, purple haze.
The camera pulls back to reveal two infinite rows of buses stretching into a foggy, dark background.
"Not just one or two, but an infinite number of infinite buses."
Setting: An abstract, foggy space — Dim light from the bus windows.
An animation shows a grid being drawn on a piece of textured, off-white paper.
"So, what can you do? Well, you pull out an infinite spreadsheet, of course."
Setting: A piece of paper — Even, flat light.
Text appears in the first column of the spreadsheet, labeling the rows for the hotel and the infinite buses.
"You make a row for each bus, bus one, bus two, bus three, and so on, and a row at the top for all the people who are already in the hotel."
Setting: A piece of paper — Even, flat light.
Text: "Hotel", "Bus 1", "Bus 2"
The spreadsheet fills with unique identifiers for each person, like 'HR1' (Hotel Room 1) and 'B1S1' (Bus 1 Seat 1). A black circle highlights 'B1S1'.
"So each person gets a unique identifier, which is a combination of their vehicle and their position in it."
Setting: A piece of paper — Even, flat light.
Text: "HR1, HR2, B1S1, B1S2, B2S1, B2S2..."
A red line is drawn on the spreadsheet, starting at 'HR1' and zigzagging diagonally across the cells to 'HR2', then 'B1S1', 'B2S1', 'B1S2', and so on.
"So, how do you assign the rooms? We'll start in the top left corner and draw a line that zigzags back and forth across the spreadsheet,"
Setting: A piece of paper — Even, flat light.
Text: "HR1, HR2, B1S1, B1S2, B2S1, B2S2..."
The red zigzag line on the spreadsheet animates as if it's being pulled taut, transforming the 2D grid of names into a single 1D line.
"Then, imagine you pull on the opposite ends of this line, straightening it out."
Setting: A piece of paper — Even, flat light.
Text: "HR1, HR2, B1S1, B2S1, B1S2..."
A composite image shows the straightened line of unique IDs from the spreadsheet at the bottom, directly corresponding to the numbered hotel doors at the top.
"It's then pretty simple just to line up each person on that line with a unique room in the hotel."
Setting: Diagrammatic space — Bright, even light.
Text: "1, 2, 3, 4, 5, 6", "HR1, HR2, B1S1, B2S1, B1S2, HR3"
A brightly colored bus with a chaotic pattern of letters pulls up. Colorful, glittery blob-people get off.
"But now, a big bus pulls up, an infinite party bus with no seats."
Setting: Urban street — Bright daylight.
People (1):
• getting off bus, furry/glittery texture hair — Various colored, glittery blobs
Text: "A's and B's in a pattern"
Several infinite strings of letters (A's and B's) appear over the scene of the blob-people standing in front of the hotel.
"So, their names all consist of only two letters, A and B, but each name is infinitely long."
Setting: Urban street — Bright daylight.
People (1):
• standing, furry/glittery texture hair — Various colored, glittery blobs
Text: "ABABABBBBAAAAAABAB...", "ABBAAAAAAAAAAAAAAA...", "ABABABABABABABABAB..."
The blue manager blob speaks to a green blob guest. A speech bubble appears with the word 'Sorry'.
"But you tell him, sorry, there's no way we can fit all of you in the hotel."
Setting: Hotel lobby — Overhead spotlights.
People (2):
• sitting, Blue fur hair — Furry blue blob
• standing, Green fur hair — Furry green blob
Text: "Sorry"
An infinite spreadsheet appears, with rows labeled 'Room 1', 'Room 2', etc. An infinitely long name made of A's and B's is typed into the cell next to 'Room 1'.
"You pull out your infinite spreadsheet again and start assigning rooms to people on the bus."
Setting: A piece of paper — Even, flat light.
Text: "Room 1", "ABBAAAAAAAAAAAAAAAAAAAAAA..."
The spreadsheet is now full of infinite names. A new row is added at the top labeled 'No Room'.
"I can still write down the name of a person who doesn't yet have a room."
Setting: A piece of paper — Even, flat light.
Text: "No Room"
On the spreadsheet, the first letter of the first name ('A') is highlighted in yellow. In the 'No Room' row, the letter 'B' appears.
"The way you do it is you take the first letter of the first name and flip it from an A to a B."
Setting: A piece of paper — Even, flat light.
Text: "B"
The process continues: the second letter of the second name is highlighted and flipped, then the third of the third, and so on, creating a new name in the 'No Room' row along the diagonal.
"And you keep doing this all the way down the list."
Setting: A piece of paper — Even, flat light.
Text: "BABABBB..."
The camera moves rapidly down the warmly lit infinite hallway again, creating a motion blur.
"The number of rooms in the Hilbert Hotel is infinite, sure, but it is countably infinite,"
Setting: An infinite hotel hallway — Bright, warm lights creating a blur.
A final shot of the Hilbert Hotel exterior. The 'NO VACANCY' sign is fully lit in red.
"Some infinities are bigger than others."
Setting: Urban street — Daylight.
Text: "Hilbert Hotel", "NO VACANCY"
A pair of hands holds up a smartphone, which is playing the video we are watching.
"is that the discovery of different sized infinities sparked a line of inquiry that led directly to the invention of the device you're watching this on right now."
Setting: Studio — Bright, clean studio lighting.
People (1):
• holding phone, not visible hair — not visible
An animation of a blue digital grid landscape under a starry night sky. The Veritasium logo appears.
Setting: Digital landscape — Glowing blue light from the grid.
Text: "Veritasium - an element of truth", "Lee Redden | Paul Peijzel | Richard Sundvall | Ruslan Khroma | Juan Benet | Sam Lutfi"