Hilbert's Infinite Hotel Paradox

A thought experiment that reveals the counterintuitive nature of infinity and transforms our understanding of mathematical sets

Hilbert's paradox of the Grand Hotel is a thought experiment created by mathematician David Hilbert that illustrates the counterintuitive properties of infinite sets.

The Scenario: A Hotel with Infinite Rooms

Imagine a hotel with infinitely many rooms, numbered 1, 2, 3, and so on without end. Each room is currently occupied by exactly one guest, meaning the hotel is completely full.

The First Paradox: One More Guest

A new guest arrives at the fully occupied infinite hotel and requests a room. In any finite hotel, this would be impossible - there are no vacant rooms.

The Counterintuitive Solution:

1. Move the guest in room 1 to room 2
2. Move the guest in room 2 to room 3
3. Move the guest in room 3 to room 4
4. Continue this pattern indefinitely

Result: Room 1 becomes vacant for the new guest, despite the hotel being "full."

The Greater Challenge: Infinite New Guests

The paradox becomes more complex when a bus arrives carrying infinitely many new guests. How can an already full infinite hotel accommodate infinitely many additional guests?

The Elegant Solution

The Strategy: Move each current guest from room n to room 2n

The Process:

• Guest in room 1 → moves to room 2
• Guest in room 2 → moves to room 4
• Guest in room 3 → moves to room 6
• Guest in room n → moves to room 2n

The Result: All odd-numbered rooms (1, 3, 5, 7, ...) become vacant, providing infinitely many rooms for the infinitely many new guests.

Mathematical Implications

What This Reveals About Infinity

This paradox demonstrates several profound mathematical concepts:

1. Infinite Sets Have Different Properties

A proper subset of an infinite set can have the same cardinality as the original set

2. Countable Infinity The hotel illustrates that both the original guests and new guests represent countably infinite sets - they can be put into one-to-one correspondence with natural numbers.

The Nature of Mathematical Reality

This thought experiment raises questions about:

• Whether infinite objects can exist in physical reality
• The relationship between logical possibility and physical impossibility

Source

Wikipedia: Hilbert's Paradox of the Grand Hotel (opens in a new tab)