Hilbert's Hotel

The Hilbert's Hotel is a paradox created by German mathematician used to demonstrate a counter-intuitive property of infinite setsWolfram Mathworld Hilbert HotelUniversity of Cambridge Hilbert's hotel in a 1925 lecture "Über das Unendliche".Hilbert, David (2013), Ewald, William; Sieg, Wilfried (eds.), David Hilbert's Lectures on the Foundations of Arithmetics and Logic 1917-1933, Heidelberg: Springer-Verlag ISBN 978-3-540-20578-4 It was popularized with 's 1947 book.Gamow, George (1947). One Two Three... Infinity: Facts and Speculations of Science. New York: Viking Press. p. 17. Reprinted by Dover Publications in 1988 ISBN 978-0486256641Kragh, H. (2014). The true (?) story of Hilbert's infinite hotel. arXiv preprint arXiv:1403.0059.
All the rooms in Hilbert's hotel, which has infinitely many rooms, are occupied. Now, suppose new guests arrive. Will it be possible to find free rooms for them? Surprisingly, the answer is yes. Imagine the hotel has a countably infinite set of rooms numbered \(1, 2, 3, ...\). Then any finite number \(n\) of guests can be accommodated without evicting the current guests by moving the current guests from room \(i\) to room \(i + n\). Furthermore, a countably infinite number of guests can also be accommodated by moving the existing guests from \(i\) to \(2i\), thus freeing up the countably infinite number of rooms \(2i-1\). This paradox demonstrates that the statement "all rooms are occupied" does not imply that "there is no more space for new guests."
== Sources ==
Category:Paradoxes