Math 2.0

Mathematical logic and foundations Theorem

The rationals are countable

Builds on Countably infinite setThe integers are countableProduct of countable sets is countable

Statement

The set ℚ\mathbb{Q} of rational numbers is countably infinite.

Remarks

Proof: each rational number has a unique form p/qp/q with p∈ℤp \in \mathbb{Z}, q≥1q \ge 1 and gcd⁡(p,q)=1\gcd(p, q) = 1. The map p/q↦(p,q)p/q \mapsto (p, q) is an injection into ℤ×ℕ\mathbb{Z} \times \mathbb{N}, which is countable. And ℚ\mathbb{Q} is infinite. The rationals are dense in ℝ\mathbb{R}, but they are countable, and ℝ\mathbb{R} is not countable.

Reading path

To understand this page from the ground up, read these 23 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 functiondefinitionCountable set, definition, no evidence levelCountable setdefinitionBijective function, definition, no evidence levelBijective functiondefinitionCantor pairing function, example, no evidence levelCantor pairing functi…exampleCountably infinite set, definition, no evidence levelCountably infinite setdefinitionProduct of countable sets is countable, statement, no evidence levelProduct of countable …statementThe integers are countable, statement, no evidence levelThe integers are coun…statementThe rationals are countable, statement, no evidence levelThe rationals are cou…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. Countable set definition
  18. Bijective function definition
  19. Cantor pairing function example
  20. Countably infinite set definition
  21. Product of countable sets is countable statement
  22. The integers are countable statement
  23. The rationals are countable statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Rational_number, Countability
About this page
Tag0DS9KWFDK2
URIm2://0DS9KWFDK2
Kindstatement, theorem
WritingW0 stub
MSC03E10
StatusThis copy of the site has no build data. Run m2 status 0DS9KWFDK2.
UsesCountably infinite set, The integers are countable, Product of countable sets is countable

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