Math 2.0

Mathematical logic and foundations Definition

Cardinal number

Builds on CardinalityOrdinal number

Definition

A cardinal number is the cardinality ∣A∣|A| of some set AA. In ZFC, the von Neumann definition takes ∣A∣|A| to be the least ordinal κ\kappa such that κ\kappa is equinumerous with AA. A cardinal number is then an ordinal that is not equinumerous with any smaller ordinal.

Remarks

The finite cardinals are the natural numbers 0,1,2,…0, 1, 2, \dots. The least infinite cardinal is aleph-null, the cardinality of ℕ\mathbb{N}. Cardinals have an order, an addition, a multiplication and an exponentiation. The sum ∣A∣+∣B∣|A| + |B| is the cardinality of a disjoint union, and ∣A∣⋅∣B∣=∣A×B∣|A| \cdot |B| = |A \times B|. The power ∣B∣∣A∣|B|^{|A|} is the cardinality of the set of functions from AA to BB.

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To understand this page from the ground up, read these 29 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 relationdefinitionAntisymmetric 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-orderdefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionOrdinal number, definition, no evidence levelOrdinal numberdefinitionCardinality, definition, no evidence levelCardinalitydefinitionCardinal number, definition, no evidence levelCardinal numberdefinition
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. Antisymmetric relation definition
  14. Reflexive relation definition
  15. Transitive relation definition
  16. Function definition
  17. Preorder definition
  18. Codomain definition
  19. Injective function definition
  20. Partial order definition
  21. Surjective function definition
  22. Least element definition
  23. Total order definition
  24. Bijective function definition
  25. Well-order definition
  26. Equinumerous sets definition
  27. Ordinal number definition
  28. Cardinality definition
  29. Cardinal number definition

Where this leads

22 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cardinal_number, lead section
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