The Infinite Hotel Paradox

The Infinite Hotel Paradox illustrates how a hotel with infinitely many rooms can always accommodate more guests, even when full. Imagine a hotel with endless rooms, all occupied. If a new guest arrives, the hotel can shift each current guest from room n to room n+1, freeing up room 1 for the newcomer. Similarly, if infinitely many guests arrive simultaneously, the hotel can relocate each current guest to a room double their current one, freeing infinitely many rooms. This paradox highlights how infinity behaves differently from finite sets, allowing for endless rearrangements without congestion.