Math 2.0

Mathematical logic and foundations Axiom

Continuum hypothesis

Builds on Aleph-nullCardinality of the continuumOrder of cardinals

Statement

There is no set SS with ℵ0<∣S∣<2ℵ0\aleph_0 < |S| < 2^{\aleph_0}. Equivalently, with the axiom of choice, 2ℵ0=ℵ12^{\aleph_0} = \aleph_1.

Remarks

In words: each infinite subset of ℝ\mathbb{R} is countable or equinumerous with ℝ\mathbb{R}. The hypothesis is independent of ZFC: if ZFC is consistent, then ZFC proves neither the hypothesis nor its negation. The generalized continuum hypothesis says that 2ℵα=ℵα+12^{\aleph_\alpha} = \aleph_{\alpha + 1} for each ordinal α\alpha.

History

Georg Cantor proposed the hypothesis in 1878. It was the first of the problems that David Hilbert presented in 1900. Kurt Gödel showed in 1940 that ZFC does not refute it, and Paul Cohen showed in 1963 that ZFC does not prove it.

Reading path

To understand this page from the ground up, read these 35 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 numberdefinitionOrder of cardinals, definition, no evidence levelOrder of cardinalsdefinitionAleph-null, object, no evidence levelAleph-nullobjectCardinality of the continuum, object, no evidence levelCardinality of the co…objectContinuum hypothesis, statement, no evidence levelContinuum hypothesisstatement
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. Order of cardinals definition
  33. Aleph-null object
  34. Cardinality of the continuum object
  35. Continuum hypothesis statement

Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuum_hypothesis, lead section
About this page
Tag0MN2MER9WN
URIm2://0MN2MER9WN
Aliasesch
Kindstatement, axiom
WritingW1 stated
MSC03E50
StatusThis copy of the site has no build data. Run m2 status 0MN2MER9WN.
UsesAleph-null, Cardinality of the continuum, Order of cardinals
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