The Infinite Hotel Paradox - Jeff Dekofsky

Introduction

The Infinite Hotel Paradox, introduced by mathematician David Hilbert, challenges our understanding of infinity through a thought experiment involving a hotel with an infinite number of rooms. This tutorial will guide you through the key concepts of the paradox, illustrating how it handles scenarios that seem impossible, such as accommodating new guests when the hotel is fully booked.

Step 1: Understand the Concept of Infinity

• Definition: Infinity refers to a quantity without bound or limit. In mathematics, it is not a number but a concept used to describe something that goes on forever.
• Hilbert's Hotel: Imagine a hotel with an infinite number of rooms, numbered 1, 2, 3, and so forth. Despite being fully booked, this hotel can still accommodate more guests.

Step 2: Accommodating One New Guest

• Scenario: A new guest arrives at a fully booked Infinite Hotel.
• Solution:
  1. Move the guest in Room 1 to Room 2.
  2. Move the guest in Room 2 to Room 3.
  3. Continue this process, moving each guest from Room n to Room n+1.
• Outcome: Room 1 becomes available for the new guest, demonstrating that even a fully booked hotel can accommodate one additional guest.

Step 3: Accommodating Multiple New Guests

• Scenario: A bus arrives with 40 new guests.
• Solution:
  1. Move the guest in Room 1 to Room 41.
  2. Move the guest in Room 2 to Room 42.
  3. Continue this process, moving each guest from Room n to Room n+40.
• Outcome: Rooms 1 through 40 become available for the new guests, showcasing the hotel’s ability to accommodate infinitely more guests despite being fully booked.

Step 4: Accommodating an Infinite Number of New Guests

• Scenario: An infinite bus arrives with an infinite number of guests.
• Solution:
  1. Assign each new guest to Room 2n, where n is their position on the bus (1st guest goes to Room 2, 2nd guest to Room 4, etc.).
  2. This method uses every even-numbered room for the new guests.
• Outcome: All odd-numbered rooms remain occupied while all even-numbered rooms become available for the infinite new guests, emphasizing the counterintuitive nature of infinity.

Conclusion

The Infinite Hotel Paradox illustrates the surprising properties of infinity, showing that even a fully booked infinite hotel can accommodate more guests through simple mathematical principles. Understanding this paradox enhances our comprehension of infinity and its implications in mathematics. For further exploration, consider diving into related thought experiments or mathematical concepts that challenge our perception of quantity and space.