A hotel with infinite rooms, all full, can always take more guests

Imagine a hotel with infinitely many rooms, every single one occupied. A new guest arrives. In a normal hotel that’s a dead end — but David Hilbert designed this one in a 1924 lecture, as a teaching device to make Georg Cantor’s then-controversial theory of transfinite numbers feel intuitive rather than absurd.

The clerk simply asks the guest in room 1 to move to room 2, room 2 to room 3, and so on — everyone shifts from room n to room n+1. Because there’s no last room, nobody is stranded, and room 1 opens for the newcomer.

It gets stranger. If infinitely many new guests show up, move each current guest from room n to room 2n, freeing every odd-numbered room. Even infinitely many coaches, each holding infinitely many passengers, can be absorbed: seat the k-th passenger of coach p in room p^k (using a different prime for each coach), and no two guests ever collide. A countable union of countable sets stays countable.

The smallest infinity, written ℵ₀, contains room for endless copies of itself.

But the trick has a hard limit. Suppose a coach pulls up with one guest for every real number between 0 and 1. Now the clerk is stuck. Cantor’s diagonal argument shows that any list of real numbers must miss at least one — build a new number that differs from the first listed number in its first digit, the second in its second, and so on. The reals form a strictly bigger infinity than ℵ₀, too large for even an infinite hotel to hold.