Hilbert Hotel

The Hilbert Hotel is a thought experiment illustrating how an infinite hotel with infinitely many rooms can still accommodate additional guests, even when it’s fully occupied. Imagine a hotel with endless rooms all occupied; surprisingly, if a new guest arrives, the hotel can still find a room for them by shifting each current guest from their room \(n\) to room \(n+1\). This move frees up room 1 for the new guest, demonstrating that an infinite collection of rooms can handle infinitely many newcomers without any problem. It highlights peculiar properties of infinity, like its ability to be both full and capable of expansion simultaneously.