Hilbert's Hotel

Hilbert's Hotel is a thought experiment illustrating the counterintuitive properties of infinite sets. Imagine a hotel with infinitely many rooms, all occupied. If a new guest arrives, the hotel can still accommodate them by moving the guest in room 1 to room 2, the guest in room 2 to room 3, and so on, creating a vacancy in room 1 for the new guest. This shows that, unlike finite hotels, an infinite hotel can always accommodate more guests—even if fully booked—highlighting the unique nature of infinity in mathematics.