The Hotel That Never Fills Up

For 1 new guest: shift everyone to room n+1. For infinite new guests: move everyone to even rooms (n → 2n), freeing all odd rooms for the new arrivals. Countably infinite sets absorb countably infinite additions.

The hotel illustrates that 'countably infinite' sets can absorb any countable addition because you can always establish a one-to-one mapping between them — the formal idea of cardinality. Step 1: Starting state: rooms 1, 2, 3, 4, ... all occupied. One new guest arrives. Step 2: Solution for 1 guest: broadcast an announcement — each guest moves from their current room n to room n+1. The guest in room 1 moves to room 2, room 2 to room 3, and so on. Room 1 is now empty. New guest checks in. Step 3: Now a coach arrives with passengers numbered 1, 2, 3, ... stretching to infinity. Step 4: Solution for infinite guests: ask every current guest to move from room n to room 2n. Guest in room 1 → room 2; room 2 → room 4; room 3 → room 6. All even-numbered rooms are now taken. Step 5: Count the now-empty rooms: 1, 3, 5, 7, 9, ... — every odd-numbered room is free. That is infinitely many available rooms. Step 6: Coach passenger number k checks into room 2k − 1. Passenger 1 → room 1; passenger 2 → room 3; passenger 3 → room 5. Everyone is housed. Key Insight: The natural numbers can be put into perfect one-to-one correspondence with the even numbers, the odd numbers, or even the set of all rational numbers. 'Countably infinite' means exactly this: no matter how you redistribute, there is always a bijection available.