Hilbert infinite hotel

Hilbert paradox, infinite hotel paradox, Hilbert 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)