Math 2.0

Mathematical logic and foundations Proposition

The integers are countable

Builds on Countably infinite setBijective function

Statement

The set ℤ\mathbb{Z} of integers is countably infinite. The function f ⁣:ℕ→ℤf \colon \mathbb{N} \to \mathbb{Z} with f(2k)=kf(2k) = k and f(2k+1)=−(k+1)f(2k + 1) = -(k + 1) is a bijection.

Remarks

The bijection lists the integers as 0,−1,1,−2,2,…0, -1, 1, -2, 2, \dots. So ∣ℤ∣=∣ℕ∣=ℵ0|\mathbb{Z}| = |\mathbb{N}| = \aleph_0, although ℕ\mathbb{N} is a proper subset of ℤ\mathbb{Z}.

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 setdefinitionThe integers are countable, statement, no evidence levelThe integers are coun…statement
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. The integers are countable statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Countable_set, Introduction
About this page
Tag066RJJVWH9
URIm2://066RJJVWH9
Kindstatement, proposition
WritingW0 stub
MSC03E10
StatusThis copy of the site has no build data. Run m2 status 066RJJVWH9.
UsesCountably infinite set, Bijective function

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