hotelliparadoksi

Hotelliparadoksi, also known as Hilbert's paradox of the Grand Hotel, is a thought experiment that illustrates the peculiar properties of infinite sets. It describes a hypothetical hotel with an infinite number of rooms, all of which are occupied.

The paradox arises when a new guest arrives. In a finite hotel, if all rooms are full, a new guest cannot be accommodated. However, in the Grand Hotel, the manager can simply ask the guest in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on. Since there is always a next room, every existing guest can be moved to a higher numbered room, leaving room 1 vacant for the new arrival.

The paradox can be extended. If an infinite number of new guests arrive, the manager can ask the guest in room 1 to move to room 2, the guest in room 2 to move to room 4, the guest in room 3 to move to room 6, and in general, the guest in room n to move to room 2n. This leaves all the odd-numbered rooms vacant, which is an infinite number of rooms, enough to accommodate the infinite number of new guests.

This thought experiment demonstrates that an infinite set can have the same cardinality as a proper subset of itself. In simpler terms, infinity is not bound by the same rules of addition and subtraction as finite numbers. It highlights the counterintuitive nature of infinity and its mathematical implications. The paradox was popularized by mathematician David Hilbert.