Hilbert's Hotel

David Hilbert imagined a hotel with infinitely many rooms, every one occupied. A tired traveler shows up wanting a room. No problem: ask the guest in room N to move to room N+1. Room 1 opens up, nobody is left without a bed, and the hotel is still full. Infinity + 1 = infinity.

Then a bus pulls up with infinitely many new passengers. Still fine: move each guest from room N to room 2N. Now every guest sits in an even-numbered room, and all the odd rooms — infinitely many — are free for the bus. A full infinity made room for another infinity.