Hilbert's Hotel Paradox

Interactive demonstration of the famous infinite hotel thought experiment. See how an infinitely full hotel can always accommodate more guests.

Choose a Scenario

The hotel is fully occupied (every room has a guest). A new guest arrives. How can the hotel accommodate them?

Hotel Status

Room Occupancy
All rooms occupied
Infinite rooms, infinite guests

Room Assignments (showing first 12 rooms)

Original guest   New guest   Moved guest

Step-by-Step Reassignment

Guest in room n moves to room n + 1

What Is Hilbert's Hotel Paradox?

Hilbert's Hotel (or the Infinite Hotel Paradox) is a thought experiment proposed by German mathematician David Hilbert in 1924 to illustrate the counterintuitive properties of infinite sets. It demonstrates that an infinite set can be put into a one-to-one correspondence with a proper subset of itself, which is impossible for finite sets.

Imagine a hotel with infinitely many rooms, numbered 1, 2, 3, 4, and so on, with no last room. Every single room is occupied by a guest. Despite being "fully booked," this hotel can always accommodate additional guests -- even infinitely many of them.

The Three Classic Scenarios

One New Guest

Move every current guest from room n to room n+1. Room 1 is now empty for the new guest.

f(n) = n + 1

k New Guests

Move every current guest from room n to room n+k. Rooms 1 through k are now empty.

f(n) = n + k

Infinitely Many New Guests

Move every current guest from room n to room 2n. All odd-numbered rooms are now empty (infinitely many of them).

f(n) = 2n

Why Does This Work?

The key insight is that the set of natural numbers {1, 2, 3, ...} is countably infinite. A countably infinite set has the special property that it can be put into a one-to-one correspondence (bijection) with a proper subset of itself. This is actually the defining property of infinite sets, known as Dedekind-infinite sets.

For finite sets, this is impossible. A hotel with 100 rooms that is fully occupied genuinely cannot accommodate another guest without someone losing their room. But infinity does not work like a large finite number -- it has fundamentally different properties.

The Mathematics Behind It

The paradox demonstrates that the cardinality (size) of the set of natural numbers does not change when you add a finite number of elements or even another countably infinite set. This cardinality is called aleph-null (represented as the Hebrew letter aleph with subscript 0), and it satisfies:

  • Aleph-0 + 1 = Aleph-0
  • Aleph-0 + k = Aleph-0 (for any finite k)
  • Aleph-0 + Aleph-0 = Aleph-0
  • Aleph-0 x Aleph-0 = Aleph-0

Historical Context

David Hilbert (1862-1943) was one of the most influential mathematicians of the 19th and 20th centuries. He proposed this thought experiment in a 1924 lecture to make the abstract concepts of set theory and Georg Cantor's work on infinite sets more accessible and intuitive. The paradox builds upon Cantor's groundbreaking work showing that not all infinities are the same size.

Beyond the Hotel: Uncountable Infinity

What if a bus arrived with uncountably many guests (as many as there are real numbers)? In this case, even Hilbert's Hotel cannot accommodate them all. Cantor's diagonal argument proves that the real numbers are strictly "more infinite" than the natural numbers, a larger cardinality called the cardinality of the continuum.