Editing Cardinality (section)

==== Hilbert's hotel ====
{{Main|Hilbert's paradox of the Grand Hotel}}
[[File:Hilbert's Hotel.png|thumb|259x259px|Visual representation of Hilbert's hotel]]
[[Hilbert's Hotel]] is a [[thought experiment]] devised by the German mathematician [[David Hilbert]] to illustrate a counterintuitive property of infinite sets (assuming the axiom of choice), allowing them to have the same cardinality as a [[proper subset]] of themselves. The scenario begins by imagining a hotel with an infinite number of rooms, all of which are occupied. But then a new guest walks in asking for a room. The hotel accommodates by moving the occupant of room 1 to room 2, the occupant of room 2 to room 3, room three to room 4, and in general room n to room n+1. Then every guest still has a room, but room 1 opens up for the new guest.<ref name=":5">{{Cite book |last=Gamov |first=George |title=One two three... infinity |title-link=One Two Three... Infinity |publisher=Viking Press |year=1947 |language=English |lccn=62-24541}} [https://archive.org/details/OneTwoThreeInfinity_158/ Archived] on 2016-01-06</ref>

Then, the scenario continues by imagining an infinite bus of new guests seeking a room. The hotel accommodates by moving the person in room 1 to room 2, room 2 to room 4, and in general room n to room 2n. Thus all the even-numbered rooms are occupied, but all the odd-numbered rooms are vacant, leaving room for the infinite bus of new guests. The scenario continues by assuming an infinite number of these infinite busses arrives at the hotel, and showing that the hotel is still able to accommodate. Finally, an infinite bus which has a seat for every [[real number]] arrives, and the hotel is no longer able to accommodate.<ref name=":5" />