Hilbert Hotel: Infinity and Beyond

Jul 14, 2024

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.