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.