Math 2.0

Mathematical logic and foundations Theorem

The reals have the cardinality of the continuum

Builds on Cardinality of the continuumSchröder–Bernstein theoremCardinality of the power set

Statement

The set ℝ\mathbb{R} of real numbers has cardinality 2ℵ02^{\aleph_0}: ∣ℝ∣=𝔠=∣𝒫(ℕ)∣.|\mathbb{R}| = \mathfrak{c} = |\mathcal{P}(\mathbb{N})|.

Remarks

One proof uses the Schröder–Bernstein theorem. A real number gives a set of rationals, the rationals less than it, so ∣ℝ∣≤∣𝒫(ℚ)∣=2ℵ0|\mathbb{R}| \le |\mathcal{P}(\mathbb{Q})| = 2^{\aleph_0}. Binary sequences give an injection in the other direction, for example through decimal expansions with digits 00 and 11. With Cantor's theorem, this gives another proof that ℝ\mathbb{R} is uncountable.

Reading path

To understand this page from the ground up, read these 37 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 relationdefinitionPower set, definition, no evidence levelPower setdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionPreorder, definition, no evidence levelPreorderdefinitionIndicator function, definition, no evidence levelIndicator functiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionPartial order, definition, no evidence levelPartial orderdefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionLeast element, definition, no evidence levelLeast elementdefinitionTotal order, definition, no evidence levelTotal orderdefinitionBijective function, definition, no evidence levelBijective functiondefinitionWell-order, definition, no evidence levelWell-orderdefinitionCountably infinite set, definition, no evidence levelCountably infinite setdefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionSchröder–Bernstein theorem, statement, no evidence levelSchröder–Bernstein th…statementOrdinal number, definition, no evidence levelOrdinal numberdefinitionCardinality, definition, no evidence levelCardinalitydefinitionCardinal number, definition, no evidence levelCardinal numberdefinitionCardinality of the power set, statement, no evidence levelCardinality of the po…statementAleph-null, object, no evidence levelAleph-nullobjectCardinality of the continuum, object, no evidence levelCardinality of the co…objectThe reals have the cardinality of the continuum, statement, no evidence levelThe reals have the ca…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. Power set definition
  14. Antisymmetric relation definition
  15. Reflexive relation definition
  16. Transitive relation definition
  17. Function definition
  18. Preorder definition
  19. Indicator function definition
  20. Codomain definition
  21. Injective function definition
  22. Partial order definition
  23. Surjective function definition
  24. Least element definition
  25. Total order definition
  26. Bijective function definition
  27. Well-order definition
  28. Countably infinite set definition
  29. Equinumerous sets definition
  30. Schröder–Bernstein theorem statement
  31. Ordinal number definition
  32. Cardinality definition
  33. Cardinal number definition
  34. Cardinality of the power set statement
  35. Aleph-null object
  36. Cardinality of the continuum object
  37. The reals have the cardinality of the continuum statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cardinality_of_the_continuum, Cardinal equalities
About this page
Tag075PM9FH7A
URIm2://075PM9FH7A
Aliasescardinality-of-the-reals
Kindstatement, theorem
WritingW0 stub
MSC03E10
StatusThis copy of the site has no build data. Run m2 status 075PM9FH7A.
UsesCardinality of the continuum, Schröder–Bernstein theorem, Cardinality of the power set

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