Math 2.0

Mathematical logic and foundations Ordinals and cardinals Theorem

Cantor's theorem

Builds on Power setSurjective functionCardinality

Statement

Let AA be a set and let f ⁣:A→𝒫(A)f \colon A \to \mathcal{P}(A) be a function to the power set of AA. Then ff is not surjective. So ∣A∣<∣𝒫(A)∣|A| < |\mathcal{P}(A)|.

Remarks

Proof: let D={a∈A∣a∉f(a)}D = \{a \in A \mid a \notin f(a)\}. If D=f(d)D = f(d) for some d∈Ad \in A, then d∈Dd \in D if and only if d∉Dd \notin D, a contradiction. The map a↦{a}a \mapsto \{a\} is an injection, so ∣A∣≤∣𝒫(A)∣|A| \le |\mathcal{P}(A)|, and the inequality is strict. It follows that there is no largest cardinal. The proof is a form of the diagonal argument, and it is close to Russell's paradox.

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To understand this page from the ground up, read these 21 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 setdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionCardinality, definition, no evidence levelCardinalitydefinitionCantor's theorem, statement, no evidence levelCantor's theoremstatement
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. Function definition
  15. Codomain definition
  16. Injective function definition
  17. Surjective function definition
  18. Bijective function definition
  19. Equinumerous sets definition
  20. Cardinality definition
  21. Cantor's theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cantor's_theorem, lead section
About this page
Tag0WTEMGYV2C
URIm2://0WTEMGYV2C
Kindstatement, theorem
WritingW1 stated
MSC03E10
StatusThis copy of the site has no build data. Run m2 status 0WTEMGYV2C.
UsesPower set, Surjective function, Cardinality
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