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Difference between revisions of "Hilbert infinite hotel"

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''Hilbert paradox, infinite hotel paradox, Hilbert hotel''
 
''Hilbert paradox, infinite hotel paradox, Hilbert hotel''
  
 
A nice illustration of some of the simpler properties of (countably) infinite sets.
 
A nice illustration of some of the simpler properties of (countably) infinite sets.
  
An infinite hotel with rooms numbered <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h1300801.png" /> can be full and yet have a room for an additional guest. Indeed, simply shift the existing guest in room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h1300802.png" /> to room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h1300803.png" />, the one in room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h1300804.png" /> to room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h1300805.png" />, etc. (in general, the one in room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h1300806.png" /> to room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h1300807.png" />), to free room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h1300808.png" /> for the newcomer.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h1300809.png" /> to room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h13008010.png" />, the one in room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h13008011.png" /> to room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h13008012.png" />, etc. (in general, the one in room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h13008013.png" /> to room <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130080/h13008014.png" />), to free all rooms with odd numbers for the newcomers.
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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]]).
 
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]]).

Latest revision as of 21:41, 14 April 2014

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)
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