Hilbert’s Hotel Paradox 

Hilbert’s Hotel Paradox is a famous mathematical thought experiment that explains how infinite sets behave differently from finite ones. It clearly demonstrates the concept of countable infinity, first formalized by Georg Cantor and later explained by mathematician David Hilbert.

What Is Hilbert’s Hotel?

Imagine a hotel with an infinite number of rooms, numbered 1, 2, 3, and continuing forever. Every room is occupied. In a normal hotel, this would mean no new guests can enter. In Hilbert’s Hotel, however, new guests can still be accommodated using mathematical rearrangement.

This hotel represents a countably infinite set, meaning its elements can be placed in one-to-one correspondence with the natural numbers.

How Can One New Guest Be Added?

Even when all rooms are occupied, one new guest can enter:

• The guest in room 1 moves to room 2
• The guest in room 2 moves to room 3
• In general, the guest in room n moves to room n + 1

This shift frees room 1 for the new guest without removing anyone.

Adding Infinitely Many New Guests

Hilbert’s Hotel can even accept infinitely many new guests:

• Each current guest moves from room n to room 2n
• All even-numbered rooms become occupied
• All odd-numbered rooms remain empty

The infinitely many empty odd-numbered rooms are then assigned to the new guests.

Mathematical Meaning of the Paradox

This paradox illustrates key principles of set theory:

• Infinite sets can have proper subsets of the same size
• Countably infinite sets have cardinality ℵ₀ (aleph-null)
• Bijections allow infinite rearrangements without loss

The result is counterintuitive but logically correct.

Important Clarification

Hilbert’s Hotel is a mathematical concept. It does not claim that physically infinite hotels exist. It is used purely to explain abstract properties of infinity.

Suggested Diagrams

• Room shifting diagram: n → n + 1
• Room doubling diagram: n → 2n
• Odd and even room separation illustration

Educational Videos

• TED-Ed – The Infinite Hotel Paradox
• Numberphile – Hilbert’s Hotel

Trusted References

• Hilbert’s Paradox of the Grand Hotel – Wikipedia
• Aleph-null – Encyclopaedia Britannica
• The True Story of Hilbert’s Infinite Hotel – Helge Kragh (arXiv)

Recommended Books

• One Two Three… Infinity – George Gamow
• Infinity and the Mind – Rudy Rucker
• The Infinite Book – John D. Barrow
• Infinity: A Very Short Introduction – Ian Stewart

Conclusion

Hilbert’s Hotel Paradox is a powerful and precise demonstration of how infinity works in mathematics. It reveals that infinite sets follow consistent rules that differ fundamentally from finite intuition.