# Hilbert infinite hotel

A nice illustration of some of the simpler properties of (countably) infinite sets.

An infinite hotel with rooms numbered \$1,2,\ldots\$ can be full and yet have a room for an additional guest. Indeed, simply shift the existing guest in room \$1\$ to room \$2\$, the one in room \$2\$ to room \$3\$, etc. (in general, the one in room \$n\$ to room \$n+1\$), to free room \$1\$ for the newcomer.

There is also room for an infinity of new guests. Indeed, shift the existing guest in room \$1\$ to room \$2\$, the one in room \$2\$ to room \$4\$, etc. (in general, the one in room \$n\$ to room \$2n\$), to free all rooms with odd numbers for the newcomers.

These examples illustrate that an infinite set can be in bijective correspondence with a proper subset of itself. This property is sometimes taken as a definition of infinity (the Dedekind definition of infinity; see also Infinity).

#### References

 [a1] H. Hermes, W. Markwald, "Foundations of mathematics" H. Behnke (ed.) et al. (ed.) , Fundamentals of Mathematics , 1 , MIT (1986) pp. 3–88 (Edition: Third) [a2] G.W. Erickson, J.A. Fossa, "Dictionary of paradox" , Univ. Press Amer. (1998) pp. 84 [a3] L. Radhakrishna, "History, culture, excitement, and relevance of mathematics" Rept. Dept. Math. Shivaji Univ. (1982)
How to Cite This Entry:
Hilbert infinite hotel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_infinite_hotel&oldid=31696
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article