The Infinite Hotel Paradox

In a lecture given in 1924, German mathematician David Hilbert introduced the idea of the paradox of the Grand Hotel, which might help you wrap your head around the concept of infinity. (Spoiler alert: it probably won't help...that's the paradox.) In his book One Two Three... Infinity, George Gamow describes Hilbert's paradox:

Let us imagine a hotel with a finite number of rooms, and assume that all the rooms are occupied. A new guest arrives and asks for a room. "Sorry," says the proprietor, "but all the rooms are occupied." Now let us imagine a hotel with an infinite number of rooms, and all the rooms are occupied. To this hotel, too, comes a new guest and asks for a room.

"But of course!" exclaims the proprietor, and he moves the person previously occupying room N1 into room N2, the person from room N2 into room N3, the person from room N3 into room N4, and so on.... And the new customer receives room N1, which became free as the result of these transpositions.

Let us imagine now a hotel with an infinite number of rooms, all taken up, and an infinite number of new guests who come in and ask for rooms.

"Certainly, gentlemen," says the proprietor, "just wait a minute."

He moves the occupant of N1 into N2, the occupant of N2 into N4, and occupant of N3 into N6, and so on, and so on...

Now all odd-numbered rooms became free and the infinite of new guests can easily be accommodated in them.

This TED video created by Jeff Dekofsky explains that there are similar strategies for finding space in such a hotel for infinite numbers of infinite groups of people and even infinite amounts of infinite numbers of infinite groups of people (and so on, and so on...) and is very much worth watching: