Hilbert Hotel: Infinity and Beyond

Introduction

• Hilbert Hotel: A hotel with infinitely many rooms numbered 1, 2, 3, and so on.
• Key Concept: Managing an infinite number of guests and rooms.

Managing Infinite Guests

Scenario 1: One New Guest

• All rooms are initially full.
• Solution: Move each guest from room n to room n+1.
• Result: Room 1 becomes available for the new guest.

Scenario 2: Finite Number of New Guests

• Example: A bus arrives with 100 new guests.
• Solution: Move each guest from room n to room n+100.
• Result: The first 100 rooms are now available for the new guests.

Scenario 3: Infinite Number of New Guests

• An infinitely long bus arrives with infinitely many guests.
• Solution: Move each guest from room n to room 2n.
• Result: All odd-numbered rooms become available for new guests (infinite odd numbers).

Infinite Buses: A Bigger Challenge

Scenario 4: Infinite Number of Infinite Buses

• Infinite buses with infinite guests arrive.
• Solution: Use an infinite spreadsheet to assign unique IDs to each guest based on their bus and seat number.
• Method:
  • Create a zigzag line through the spreadsheet that covers each unique ID.
  • Straighten the line to match guests with rooms.

Uncountable Infinity: An Impossibility

Scenario 5: Infinite Party Bus with Infinite Names

• Guests have infinitely long names with only 'A' and 'B'.
• Problem: The number of these names (uncountably infinite) exceeds the number of hotel rooms (countably infinite).
• Explanation:
  • Even with a complete list, it's possible to generate a new name that is not on the list.
  • Method: Flip the nth letter of the nth name to create a unique name (diagonal argument).
  • Result: Some infinities (uncountable) are bigger than others (countable).

Key Takeaways

• Understanding countably infinite vs. uncountably infinite.
• Some infinities are larger and more complex to manage than others.
• This concept influenced the development of modern computational devices.