How An Infinite Hotel Ran Out Of Room

Summary by Krater

This video explores Hilbert's Grand Hotel paradox, demonstrating how an infinite hotel can always accommodate new guests, even when presented with an infinite number of guests or an uncountably infinite number of guests on buses.

From the video

Answers: How does Hilbert's Grand Hotel paradox work with infinite guests and infinite buses?

• Hilbert's Grand Hotel paradox
• infinity in mathematics
• countable infinity
• uncountable infinity
• Cantor's diagonal argument

What it concludes

• A hotel with an infinite number of rooms can accommodate a new guest by shifting every existing guest from room n to room n plus one.
• A hotel with an infinite number of rooms can accommodate an infinite number of buses carrying infinite guests by moving each guest to double their room number and placing bus passengers in the odd-numbered rooms.
• Some infinities are bigger than others, meaning uncountably infinite groups of guests cannot be accommodated in a countably infinite hotel.