Math 2.0

Mathematical logic and foundations Theorem

Comparability of cardinals

Builds on Order of cardinalsAxiom of choice

Statement

Assume the axiom of choice. Let AA and BB be sets. Then there is an injective function from AA to BB or an injective function from BB to AA. So ∣A∣≤∣B∣|A| \le |B| or ∣B∣≤∣A∣|B| \le |A|.

Remarks

With the Schröder–Bernstein theorem, it follows that the order of cardinals is a total order. The statement is equivalent to the axiom of choice over ZF. One proof uses Zorn's lemma on the set of injective partial functions from AA to BB.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionChoice function, definition, no evidence levelChoice functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionAxiom of choice, statement, no evidence levelAxiom of choicestatementBijective function, definition, no evidence levelBijective functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionCardinality, definition, no evidence levelCardinalitydefinitionOrder of cardinals, definition, no evidence levelOrder of cardinalsdefinitionComparability of cardinals, statement, no evidence levelComparability of card…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. Disjunction definition
  5. Implication definition
  6. Predicate definition
  7. Family of sets definition
  8. Ordered pair definition
  9. Union definition
  10. Existential quantifier definition
  11. Universal quantifier definition
  12. Cartesian product definition
  13. Subset definition
  14. Union of a family of sets definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Codomain definition
  19. Injective function definition
  20. Choice function definition
  21. Surjective function definition
  22. Axiom of choice statement
  23. Bijective function definition
  24. Equinumerous sets definition
  25. Cardinality definition
  26. Order of cardinals definition
  27. Comparability of cardinals statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cardinal_number, Cardinal comparison
About this page
Tag0YBHSJMN3J
URIm2://0YBHSJMN3J
Aliasestrichotomy-of-cardinals
Kindstatement, theorem
WritingW0 stub
MSC03E10
StatusThis copy of the site has no build data. Run m2 status 0YBHSJMN3J.
UsesOrder of cardinals, Axiom of choice

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