#MathsInMinutes Day 25: Hilbert´s hotel

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Hilbert´s hotel is an analogy invented by mathematician David Hilbert in order to visualize the strange idea of countable infinities. This imaginary hotel has a countably infinite set of rooms numbered 1, 2, 3, …, and is fully occupied, when a latecomer arrives and pleads for a room.

After some thought, the concierge uses a Tannoy system to ask every guest to move into the next room up in numerical order. So the occupant of room 1 moves into room 2, room 2 moves to room 3 and so on. For any of the, countably infinite, guests in room N, there is always a room N+1 for them to move into, so that by the time everyone has moved, room 1 is free for the new guest to occupy.

Hilbert´s hotel shows that the result of adding an element to a countably infinite set is a still countably infinite set, so there must be different countable infinites.

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