Let a hotel have a denumerable set of rooms numbered 1, 2, 3, .... Then any finite number 
of guests can be accommodated without evicting the current
guests by moving the current guests from room 
to room 
. Furthermore, a denumerable
number of guests can be similarly accommodated by moving the existing guests from
to 
, freeing up the denumerable
number of rooms 
.
Hilbert Hotel
See also
Cardinal Number, Denumerable SetExplore with Wolfram|Alpha
References
Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 84-85, 1998.Fadiman, C. Fantasia Mathematica, Being a Set of Stories, Together with a Group of Oddments and Diversions, All Drawn from the Universe of Mathematics. New York: Simon and Schuster, p. 286, 1958.Gamow, G. One, Two, Three, ... Infinity. New York: Dover, 1988.Hilbert, D. "Lectures on the Infinite." Ch. 4 in David Hilbert's Lectures on the Foundations of Arithmetic and Logic 1917-1933 (Ed. W. Ewald and W. Sieg). New York: Springer, pp. 668-760, 2013.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, p. 222, 1998.Lauwerier, H. "Hilbert Hotel." In Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, p. 22, 1991.Pappas, T. "Hotel Infinity." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 37, 1989.Referenced on Wolfram|Alpha
Hilbert HotelCite this as:
Weisstein, Eric W. "Hilbert Hotel." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HilbertHotel.html