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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.
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