Math 2.0

Mathematical logic and foundations Example

Hilbert's hotel

Builds on Countably infinite setInjective function

Example

A hotel has rooms numbered 1,2,3,…1, 2, 3, \dots, one for each positive integer, and each room has a guest. A new guest arrives. The manager moves the guest of room nn to room n+1n + 1 for each nn. Then room 11 is free for the new guest. Next, infinitely many new guests g1,g2,…g_1, g_2, \dots arrive. The manager moves the guest of room nn to room 2n2n, and gives room 2k−12k - 1 to gkg_k. In both cases, each guest has a room and no two guests share a room.

Remarks

The maps n↦n+1n \mapsto n + 1 and n↦2nn \mapsto 2n are injections of ℕ>0\mathbb{N}_{>0} into a proper subset of itself. So the set of rooms is equinumerous with a proper subset. This cannot happen for a finite set. David Hilbert introduced the hotel in a lecture of 1924 to 1925, and George Gamow made it popular in a book of 1947.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionCountably infinite set, definition, no evidence levelCountably infinite setdefinitionHilbert's hotel, example, no evidence levelHilbert's hotelexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Existential quantifier definition
  8. Universal quantifier definition
  9. Cartesian product definition
  10. Subset definition
  11. Uniqueness quantifier definition
  12. Binary relation definition
  13. Function definition
  14. Codomain definition
  15. Injective function definition
  16. Surjective function definition
  17. Bijective function definition
  18. Countably infinite set definition
  19. Hilbert's hotel example

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel, lead section
About this page
Tag0N3Y8G340X
URIm2://0N3Y8G340X
Aliaseshilbert-grand-hotel, hilberts-hotel
Kindexample
WritingW2 usable
MSC03E10
Instance ofCountably infinite set
UsesCountably infinite set, Injective function
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