Hilbert's Hotel

The hotel and its operations

The hotel has infinitely many rooms, numbered 1, 2, 3, and so on without end. Every room is occupied. A new guest arrives at the front desk.

No problem. The manager announces over the intercom that every current guest should move to the room with the next number: the guest in room 1 moves to room 2, room 2 to room 3, and so on. This is possible because there is no last room. Room 1 is now empty. The new guest checks in.

An infinite bus arrives, carrying infinitely many new guests. The manager asks all current guests to move to the room with double their current number: room 1 to room 2, room 2 to room 4, room 3 to room 6. Every odd-numbered room is now empty. There are infinitely many odd numbers. Every new guest gets a room.

What it reveals about infinity

In finite arithmetic, "full" means no capacity remains. Hilbert's Hotel shows that for infinite sets, this intuition breaks down. Adding one to infinity, or even adding infinity to infinity, does not produce a "larger" infinity in any meaningful sense.

Georg Cantor's work provides the mathematics behind this. Cantor showed that infinite sets can be put into one-to-one correspondence with proper subsets of themselves. The natural numbers can be matched, one for one, with the even natural numbers, even though the evens seem like "half" of the naturals. This is the defining property of infinite sets: they are, in a precise sense, as large as their own proper subsets.

Cantor also showed that not all infinities are equal. His diagonal argument proves that the real numbers cannot be put into one-to-one correspondence with the natural numbers. The infinity of real numbers is strictly larger. There is a hierarchy of infinities, and the hotel only demonstrates the smallest kind.

Can actual infinities exist?

The hotel is mathematically coherent, but can anything like it exist in the physical world? This is a live philosophical debate. Most cosmologists and philosophers of physics are skeptical that actual infinities, as opposed to potential infinities (limits that can be approached but never reached), are physically realized.

Aristotle distinguished between the two: a line can be divided without end, but there is no actually infinite division. Many physicists suspect that spacetime is discrete at the Planck scale, ruling out the infinite divisibility that would require actual infinities. If they're right, Hilbert's Hotel is a mathematical object without a physical home. If they're wrong, the universe may be stranger than the hotel already suggests.