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''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|Infinity]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Hermes,   W. Markwald,   "Foundations of mathematics"  H. Behnke (ed.)  et al. (ed.) , ''Fundamentals of Mathematics'' , '''1''' , MIT  (1986)  pp. 3–88  (Edition: Third)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.W. Erickson,   J.A. Fossa,   "Dictionary of paradox" , Univ. Press Amer.  (1998)  pp. 84</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Radhakrishna,   "History, culture, excitement, and relevance of mathematics"  ''Rept. Dept. Math. Shivaji Univ.''  (1982)</TD></TR></table>