Hilbert's Paradox

Hilbert’s Paradox, often called Hilbert’s Hotel, illustrates a fascinating idea about infinite sets. Imagine a hotel with infinitely many rooms, all occupied. Surprisingly, if a new guest arrives, the hotel can still accommodate them by moving each current guest from their room n to room n+1, freeing up room 1. This shows that in an infinite hotel, adding even one guest doesn’t cause chaos—you can always find a way to make room. The paradox highlights the counterintuitive nature of infinity, demonstrating that infinite sets behave differently from finite ones in terms of capacity and arrangement.