How An Infinite Hotel Ran Out Of Room

Summary

The Hilbert Hotel, a thought experiment, explores the concept of infinity by presenting a hotel with an infinite number of rooms. Initially, the hotel accommodates guests in a one-to-one room arrangement. When faced with new arrivals, the manager ingeniously reassigns rooms to accommodate them. However, when confronted with an infinite number of infinite buses, the manager uses a systematic approach to assign rooms, illustrating different sizes of infinity. The script concludes by contrasting countable infinity of the hotel's rooms with the uncountable infinity of the people on the bus, highlighting that some infinities are indeed larger than others.

Takeaways

• 🏨 The Hilbert Hotel is a thought experiment featuring an infinite number of rooms, each occupied by a guest.
• 🔄 Even with all rooms occupied, a new guest can be accommodated by moving everyone down one room.
• 🚌 When a bus with 100 guests arrives, the manager can shift all guests down 100 rooms to make space.
• ♾️ The hotel can accommodate an infinite number of new guests by moving each existing guest to a room with double their current room number.
• 📈 The hotel can even handle infinite buses with infinite guests by organizing them using a unique identifier system.
• 📊 By mapping guests from an infinite grid to a single infinite line, everyone can be assigned a room.
• 🔡 A new challenge arises when an infinite party bus arrives with guests having infinitely long names composed of two letters, A and B.
• ❌ It's impossible to accommodate everyone from the party bus, as their number is uncountably infinite, unlike the hotel’s countably infinite rooms.
• 🚫 Even with a complete list of guests, it's possible to construct a name that does not appear on the list, illustrating the concept of uncountable infinity.
• 🤯 The discovery of different sizes of infinity has profound implications, including influencing the development of modern computing.

Q & A

What is the Hilbert Hotel?

The Hilbert Hotel is a hypothetical hotel with an infinite number of rooms, each numbered consecutively, used to illustrate various paradoxes and concepts related to infinity.

How does the manager accommodate a new guest when all rooms are occupied?

When all rooms are occupied and a new guest arrives, the manager instructs all current guests to move to the room number that is double their current room number, thus freeing up all the odd-numbered rooms for the new guest.

What happens when a bus with an infinite number of people arrives?

When a bus with an infinite number of people arrives, the manager has each guest move to a room number that is double their current room number, which leaves an infinite number of odd-numbered rooms available for the new guests.

How does the manager handle an infinite number of infinite buses arriving at the hotel?

The manager creates an infinite spreadsheet with rows for each bus and columns for each person's position. By zigzagging across this grid, a unique identifier is created for each person, allowing them to be matched with a room.

What is the significance of the party bus with people identified by unique infinite sequences of 'A' and 'B'?

The party bus represents a scenario with an uncountably infinite number of people, which is a different order of infinity compared to the countably infinite number of rooms in the Hilbert Hotel. This illustrates that not all infinities are the same size.

Why can't the Hilbert Hotel accommodate the people from the party bus with unique infinite sequences?

The Hilbert Hotel cannot accommodate all the people from the party bus because the number of unique infinite sequences of 'A' and 'B' is uncountably infinite, which is a larger infinity than the countable infinity of the hotel's rooms.

What does it mean for infinities to be 'countable' or 'uncountable'?

A countable infinity is one that can be matched one-to-one with the set of natural numbers, like the rooms in the Hilbert Hotel. An uncountable infinity is larger and cannot be matched one-to-one with the natural numbers, like the number of unique infinite sequences of 'A' and 'B'.

How does the manager demonstrate that there are more people on the party bus than rooms in the hotel?

The manager demonstrates this by showing that for any list of infinite sequences of 'A' and 'B', it's possible to create a new sequence that differs from each on the list, indicating that the set of sequences is uncountably infinite.

What is the relevance of the different sizes of infinities to the invention of modern technology?

The understanding of different sizes of infinities has been foundational in the development of set theory and mathematical logic, which are crucial to the development of computer science and technology, including the devices used to watch this video.

What is the 'diagonal argument' mentioned in the script?

The diagonal argument is a proof technique used to show that the set of all possible infinite sequences of 'A' and 'B' is uncountable. It involves creating a new sequence that differs from each sequence on a hypothetical list by flipping alternate letters.

Outlines

00:00

🏨 The Paradox of Infinite Rooms: Managing Hilbert's Hotel

The Hilbert Hotel, a thought experiment, features an infinite number of rooms, all filled with guests. When new guests arrive, the manager cleverly reassigns rooms, shifting each guest down one or more rooms to make space. However, the true challenge emerges when an infinite number of new guests arrive. By using mathematical creativity, the manager assigns them to available odd-numbered rooms. Yet, when infinite buses arrive with infinite guests, the manager uses a zigzag pattern on a spreadsheet to assign rooms. The process works until guests with infinite, unique, and non-repeating names arrive, illustrating the limits of the hotel's capacity and the fascinating nature of infinity.

05:01

🔢 Different Sizes of Infinity: The Uncountable vs. Countable Infinities

Although Hilbert's Hotel has an infinite number of rooms, the infinity it represents is countable, meaning it aligns with the positive integers. However, the guests from the infinite buses represent an uncountable infinity, a larger, more complex infinity that cannot be matched one-to-one with the rooms. This uncountable infinity leads to the realization that some infinities are bigger than others. The concept is mind-boggling, and it directly connects to mathematical developments that eventually led to modern computing. This exploration of infinity reveals that even within infinity, there are limits.

Keywords

💡Hilbert Hotel

The Hilbert Hotel is a thought experiment in mathematics, used to illustrate the concept of infinity. It is a hotel with an infinite number of rooms, each numbered consecutively. In the video, the hotel is used to explore how one might accommodate new guests even when all rooms are supposedly occupied, showcasing the surprising properties of infinity. For instance, when a new guest arrives and all rooms are full, the manager instructs the current guests to move to the next room, thus creating a vacancy in room one for the new guest.

💡Infinity

Infinity is the concept of something being boundless or endless. It is a fundamental idea in mathematics, particularly in set theory and calculus. The video uses the Hilbert Hotel to demonstrate different types of infinities and how they can be manipulated. For example, the hotel can accommodate an infinite number of guests because the infinity of rooms can be rearranged to make space for new arrivals.

💡Countably Infinite

Countably infinite refers to a set of items that can be paired one-to-one with the set of natural numbers, meaning they can be counted. In the script, the rooms in the Hilbert Hotel are described as being countably infinite, as there is a one-to-one correspondence with the set of all positive integers. This concept is used to explain how the hotel can accommodate an infinite number of guests by having them move to rooms with double their current room number.

💡Uncountably Infinite

Uncountably infinite is a concept in mathematics where a set is too large to be put into one-to-one correspondence with the set of natural numbers. This type of infinity is illustrated in the video with the arrival of an infinite party bus where each person has a unique, infinitely long name composed of 'A's and 'B's. The video explains that the number of such names is uncountably infinite, which exceeds the countable infinity of the hotel rooms.

💡Zigzagging Assignment

In the context of the video, zigzagging assignment refers to the method used to assign rooms to an infinite number of infinite buses. The manager creates an infinite spreadsheet and assigns rooms by drawing a line that zigzags across the grid, ensuring each person gets a unique room. This method is a creative way to visualize how to handle an uncountable infinity within a countable one.

💡Diagonalization

Diagonalization is a technique used in set theory to prove that certain sets are uncountably infinite. In the video, the manager uses diagonalization to show that even if all rooms are assigned to guests with unique names, it's possible to create a new name that hasn't been assigned a room. This is done by flipping the first letter of the first name, the second letter of the second name, and so on, creating a name that differs from every name on the list.

💡Spreadsheet

In the video, the spreadsheet is a metaphorical tool used to organize and assign rooms to the infinite number of guests. It represents a systematic approach to handling infinities, where each row could represent a bus and each column a room. The spreadsheet is used to demonstrate the concept of mapping and the limitations of countable infinity when faced with uncountable infinity.

💡Room Numbering

Room numbering in the video is a critical element in the thought experiment. It starts with a simple one-to-one correspondence with natural numbers but evolves to include doubling room numbers and other patterns to accommodate more guests. This concept is central to understanding how the hotel can seemingly defy the limitations of a finite space by creatively renumbering and reallocating rooms.

💡Guests

Guests in the video represent the elements of infinite sets. The script explores how these guests, whether finite or infinite in number, can be accommodated in the Hilbert Hotel. The concept is used to explain different types of infinities and the challenges in managing them, such as when an infinite number of infinite buses arrive, each with its own set of guests.

💡Bus

In the video, buses symbolize the arrival of new, potentially infinite sets of guests. Each bus represents a different scenario that tests the manager's ability to accommodate an infinite number of guests. The buses carrying infinite people illustrate the concept of uncountable infinity, challenging the manager to find a way to fit them into the hotel's countably infinite rooms.

💡Names

Names in the video, particularly those consisting of infinite sequences of 'A's and 'B's, are used to represent uncountable infinities. The script uses this concept to demonstrate that there are more names than there are rooms in the Hilbert Hotel, even though both are infinite. This illustrates the idea that not all infinities are the same size and that some are larger than others.

Highlights

Introduction to the concept of the Hilbert Hotel, an infinite hotel with an infinite number of rooms.

The Hilbert Hotel is always full, yet it can still accommodate new guests by shifting current occupants.

The process of moving guests to double their current room number to make room for infinitely many new guests.

Explanation of countably infinite sets, such as the rooms in the Hilbert Hotel and the infinite set of odd numbers.

Introduction of the concept of uncountably infinite sets, with an infinite number of guests on the bus identified by infinite sequences of two letters, A and B.

Demonstration of why the uncountably infinite set of guests cannot be accommodated in the countably infinite rooms of the Hilbert Hotel.

Use of a diagonalization argument to show that there's always a name that can't be assigned a room.

Distinction between countable and uncountable infinities, showing that some infinities are larger than others.

The paradoxical nature of infinity and the limits even within infinite sets.

The notion that infinity can be expanded or manipulated, but still has inherent limitations depending on its nature.

The realization that despite having infinite resources (like rooms), some infinities (like guests) are too large to be accommodated.

The idea that the Hilbert Hotel represents a fundamental concept in mathematics regarding the nature of infinity.

The explanation of how these ideas about infinity relate to modern computing.

The transcript ends with a hint that the discovery of different-sized infinities has influenced the invention of modern technology.

The overarching theme that infinity, while seemingly limitless, has complexities and hierarchies that challenge our understanding.

Transcripts

- 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 let's 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 1 bus 2 bus 3 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, et cetera.

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?

Well 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, A, A,

and so on forever.

Someone else is named AB, AB, AB, AB, AB, et cetera.

On this bus, there's a person with

every possible infinite sequence of these two letters.

Now, ABB, 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 AB AB AB AB repeating.

And you keep going, putting a different string

of As and Bs 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 leftover.

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.

(upbeat music)