Math 2.0

Mathematical logic and foundations Object

Cardinality of the continuum

Builds on Cardinal numberPower setAleph-null

Description

The cardinality of the continuum, written 𝔠\mathfrak{c}, is the cardinal number 2ℵ02^{\aleph_0}. It is the cardinality of the power set of ℕ\mathbb{N}, and of the set of all sequences of 00 and 11. It is also the cardinality of ℝ\mathbb{R}. See the reals have the cardinality of the continuum.

Remarks

By Cantor's theorem, 𝔠>ℵ0\mathfrak{c} > \aleph_0. Other sets of cardinality 𝔠\mathfrak{c} are each nonempty open interval of ℝ\mathbb{R}, the plane ℝ2\mathbb{R}^2 and the set of irrational numbers. The continuum hypothesis says that 𝔠=ℵ1\mathfrak{c} = \aleph_1.

Reading path

To understand this page from the ground up, read these 33 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 levelPreorderdefinitionCodomain, 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 setsdefinitionOrdinal number, definition, no evidence levelOrdinal numberdefinitionCardinality, definition, no evidence levelCardinalitydefinitionCardinal number, definition, no evidence levelCardinal numberdefinitionAleph-null, object, no evidence levelAleph-nullobjectCardinality of the continuum, object, no evidence levelCardinality of the co…object
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. Codomain definition
  20. Injective function definition
  21. Partial order definition
  22. Surjective function definition
  23. Least element definition
  24. Total order definition
  25. Bijective function definition
  26. Well-order definition
  27. Countably infinite set definition
  28. Equinumerous sets definition
  29. Ordinal number definition
  30. Cardinality definition
  31. Cardinal number definition
  32. Aleph-null object
  33. Cardinality of the continuum object

Where this leads

12 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cardinality_of_the_continuum, lead section
About this page
Tag0WC28W9DNM
URIm2://0WC28W9DNM
Aliasescontinuum-cardinality
Kindobject
WritingW1 stated
MSC03E10
Instance ofCardinal number
UsesCardinal number, Power set, Aleph-null
Same as

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What this page needs for W2

  • section Invariants: a table of the main invariants (0 of 1)

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