Math 2.0

Mathematical logic and foundations Object

Aleph-null

Builds on Cardinal numberCountably infinite set

Description

Aleph-null, written ℵ0\aleph_0, is the cardinality of the set ℕ\mathbb{N} of natural numbers. It is the cardinality of each countably infinite set. It is the least infinite cardinal number: each infinite set has cardinality at least ℵ0\aleph_0, with the axiom of choice.

Remarks

Other names are aleph-zero and aleph-nought. ℵ0+ℵ0=ℵ0⋅ℵ0=ℵ0\aleph_0 + \aleph_0 = \aleph_0 \cdot \aleph_0 = \aleph_0, but 2ℵ0>ℵ02^{\aleph_0} > \aleph_0 by Cantor's theorem. As an ordinal, ℵ0\aleph_0 is ω\omega, the set of the finite ordinals.

Reading path

To understand this page from the ground up, read these 31 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-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-nullobject
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. Countably infinite set definition
  27. Equinumerous sets definition
  28. Ordinal number definition
  29. Cardinality definition
  30. Cardinal number definition
  31. Aleph-null object

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Aleph_number, Aleph-zero
About this page
Tag0V2H5PGJ9B
URIm2://0V2H5PGJ9B
Aliasesaleph-zero, aleph-nought
Kindobject
WritingW1 stated
MSC03E10
Instance ofCardinal number
UsesCardinal number, Countably infinite set
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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